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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: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueXX 5000816 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b6a18979f258b8c6ed3 ccafbc21aca420c9c83eea371adc20e038b4d7b8ac303004b0aa205f04135140 76407216032fdd22e6219da8f16b28ca12524deb7bca073cc5eba65c102a5e85 fd48e6d062cd4283ee570a7774597e5bf0e3400b6be72db0115f3cb12db70ce0 83722870cddfadee715f10f1fcaf20e06f3c54afe5ca238539bfe2b596116e83 f5371ff18fa5003d8543226cfd4025f9940365b392a858d27f078d3abcffe4a1 54e78c7692d1a32bf935967c64f01b24788ff8325d61145e2d4a489fd986fb77 38e6b254522c77ca2797a504a9ce4676a77ebacb026eca94dde5922c936f8e90 c43e2851973f31a3280c08220536dd2c2de1ffd15fc739091fb7cd5fb9bba2be a5ab0785843300a48f080377b45809107ea6c4e3b0bf4e928ef65d468a19dc1d 34a8be67669f93aa7ca73e2eda716c26be7f59bd96bede57fab460fb9288c1fa 65e8a9b4c88ac5e57507c95331aacbfe1f62fba87b107c820043108dcccfe37e 17de40413b4fac769ccca6810fe903e08efed51303c3db661045b4d6aee2089b 422251d6f63c3ad004504aac463be1e4ea5cf2b048263022c093e14d7f7dab44 3a3da51b53f5e6bced4ea82fa22dd3633e9e6069d0694613a54583bdcc2b1a21 e897954c25820fe0a7f7faf279136b6924b058da2386836ae8ebfeeb572031ef 7a6bf1d70626646d1551d010f8d4f5b731dc8af9419b97c43470cb339f8acc51 943fb46a95a2793d35214a2f8c5d368929ed285887b8d5123d4327d780681c7d becd2fa817f17841fd8ed983eae1eff9b938ca438dd417c26bcf382baf2d779d 0033268eca572929ae82d14143d4a7542be302cc6ab1a7780a66509fb8e2d8da bb7d71d73f972c84a0b6f1e9db85690b8e865f89baf6d5babb6c0dd5127a1cf9 f129176458c7bf21bd294f3c16391d6ccfe3c24a48998f 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 54 /six put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueXX 5000789 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486858b2a4b5a0cc2e1bf4e2a4a0e748483c3bcf 5de47cc5260a3a967cac70a7a35b88b54315191d0423b4065c7a432987938c6b edad3b72ad63c2918b6e5a2017457e0d4ebc204b031f3fc6c13d7da7277a94ba 018e9998b3dd888011a5d7c4204989f30f908b95533bda845746b673ab71ea57 65a0d14f4350707e47c8276305b28513cbe1bb0dbd269a53719bda46e536685d df78ca0146b6b93e760256b74d939d4e35b5e77238f04c92298dfdd188feea30 e053eefbcbb52f2011772b3aae39f5805597bbc1e8bb75a446ce014030f4f2f0 f49f9e962ee4a1024a746fa92a3628db5270732b54e43fe5ecfa524f127e5fcc 788e77e66098336ad67fe4cccaf0253272d5df79864bf4b734cb9a5859d557d8 bc11b8e00221ebc12e97de4b1f466ead83a4c894709363bca9040410a52d592e 34ee40cc7e5efa920546b981aa659513a24b1b85c221a1875b62d0b89e57a368 321b8043a5b094e0379760a443d632892b14ad6d19dacc8c78093243ad67e6a3 08e56e6b68412ee690b10dac6e17708754a00d51fc957b500eb80175716eef4b 2ca1ef867614659bee3f2b7319e97b6fdf1efc847bf3cee3156f72f21751da8e 5fb6898919e6799820d3de0642d756e09d6fae4ff08dd3deda3173bff4bb11f7 9109c97ddc05897af709ea199a90fcee8ce4c7a3c15b18170c41c04de2d3fba8 f34296a95b8e1e8de3739b17273f8f2c85e914615e8eac5e8bd2387ba3b1edf4 7968f06e2067d836d0f9f3e085cdfd2de06a62c81d786b304326f7002e83160a 36598589228b4dddddc43c85e1d126f8fe81b828028e26317af5894aaccf4f69 6301e1a9fc45935d8a414957f08febebbc3a72ada80f101e47447d019ade56e9 f4fab969bba2b44e47399fedf5caa1bcea216d7ba713d523de97a57cbed59a39 c235a135d115ebd4abbd5c9615ae06b3a137bd5ee4181428352e4ac5a2711b30 6772a97888d406f3e2c874ec8fdad2fe950aea02869a0272574daa4451cf537b 7e4309bfd807ea34c2e6913bd02662ff2c1bdac64da947e4dcf6894c3df41ba6 58257473f2aa05c0c5584e7fd244c8382fc89050897673bc721ad94f71400579 256f659bd09afd2a6f90e291dabacf9fa81dffd96d3d140248f631160e000028 7d2e10b62016e54b7f5012a54caf5738c007d61fd9428e0bd64d1dbb20c2ebd6 bc72291ddc277691b65f21c8e9b37a26be03a2071f919dc0afaebba116280b98 43f40c10da6ab0d3443f0a126dd2f05cb0dce34e386c6561c62b6348d1178595 be6a4d612c0c147d855511d97cca6de246cc5283911f8d701f9136b06afdd794 9ec04d905d18f0ea3d7a4342e92c20c1dbc804d9a9ae2859519ae20eb86239ff 71147fe4b385684b165388798ddb915285534092928b7368d467acb5c01493d6 f5c0d36d207943b9566035572232eb2d8aa007c0aa9ec42002769ab5a69ab661 9d78816d0f8eb5d9423c43e6d1c1818e2a03631ad91275b2cdaabfbf9f259f2a 855d142c7ad14cfd02b1d30cc0950eb68a5f58f2e83e9fbdc40a2d56393a562b 98dfe628838d45d6773d3966dd91db6f3e2e4a849403def07e3bfc306877d6a1 28d0540193 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 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 66 /parenleftex put dup 67 /parenrightex put dup 104 /bracketleftBig put dup 105 /bracketrightBig put dup 110 /braceleftBig put dup 111 /bracerightBig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueXX 5000774 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d5649664203ab24acb938d58d246707ffe7 d62f04bec4b70c21ef75beb2b812622b3c74e969d72d3cd11bd7106294a99caf 0b1629bc7d4de6b96ca82930831d64575f23f4ad06a0e45e315b1d392411be8d 6d73c998789ff258a07a3c8c2057325784514c845500bfd1a971310cfc11d41c 1a167dbd5ff012c60add4e87325f6e5299032a839de65fb1473a166aae1876a4 414a434f22c1d241591fb36f857df6fa930608750ffc0c54f44994662b1f00f1 400bf752ea8d83ffc4cb77a290bc2d99981ae59a191748ba5c7ba1a9d2583fd2 1398452b6ff5d83a059f7eadcd2ef744e9dd22bdf9c79d049bf06835e878c32b 7765c69bdd8ef4deb4ea7cfff4cf9354a4ddffa689de961d16772491c7afbd7f ffde42400764c68e954ee5c455a5687959829bc3b319b2147deaab3628662c80 30c5e02fea09609abe4eaa12e217bc3af673f1bc36a7039eb13fcacb4218fe0f c5a3f9452d4edf46cc91db67b624d4f2d37502fb9f11af4da18ca40b61097f95 d44329375467ed13c5cb585ec53f62b83ef9502cc755af44bf32b87b8ae9f3f2 f8dbf72dab90acafbacd280db6aaffaefdff6d5eff26669bac56280a950560e3 d01714ba38659b0e4e28bf8cdc1f15ff7e3ea9d95a6afbbe3d8e6905aa6d531d 6b6dcab89e2036cc492c65f76ddcc3cfcff33fde3d9ffd10f7a90530fd5c7683 4329baa902e701ca42717476298c9ff511b1941244bb1dfd0e5cab92a43970cd 1bdb292eadd055c97972844fdcdb0aa06401e7b2632b9d134f96bce0064860f3 9859da7a9a82be0a840672a1331212752ae1b521b693b384a72dba8d37a86eaf a0341e7fc5f6ef7a8c566aacb91e8bf5005c18837894c4eacff41dba54e071ca 83fd01c81afd0f8ed01e8e7e1f83d1ca84a84329730700231dae848ef29b5b3a e44e85904c011443ecd1e05c64134e5f9a0bf1d9306c0e41fccc29d85e2c71dd ab46d2b64e97464b7d5278d61a5cb1f93e28e0737293f6ecbc15f1792d8e6fb1 0c18289a8e97cccb78741e68e41109cade7ab7f93ba68544a49f5e116fa186b5 635da5b39489a106cd9e733cc43f7a9b9622c9fe393ade1b1ff6be6fb3f0341b c23eac7148ff08629e55b1bdbff4074bcf653d6a3e789d2b799dc06204794723 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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 14 /delta put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 26 /rho put dup 33 /omega put dup 59 /comma put dup 65 /A put dup 77 /M put dup 78 /N put dup 79 /O put dup 81 /Q put dup 85 /U put dup 88 /X put dup 98 /b put dup 99 /c put dup 100 /d put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 114 /r put dup 121 /y put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueXX 5087383 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 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3424eb0dd282aec7c335a4a0faa558de71db32833bf5d36bcc32b938d16b72a6 4b3eeb166dac4a53ddee10e96d60a9e57a704fffde1f5b2a84b7940b46277144 63124634b7ebe2b2cca2197e7d61964cd298a39fcdafe4821cc5e3e13621a082 9747785e87709e7fd674dff769777377aa7a555d6410b52da764c9f62c3c427c b16feae2faf103eec6c7b1f05d8dccfa3896f4edb4836b9b8cf2c6f104e41b52 877eccefc240a223c91a9eeaa4170def2dd80c689fe844c7aff99488d4ed7417 f54a6963ceba142797a09f170ee8125933ea1e2b6204ce370b6f1a0938983d59 922251744d3d8ee113553fd9262a96ad48113b646e1be8343568f8568817ef4b 9835a7074da555c03247ba9e327fa8197165eb953f19fe3c44d9e5f7a97adf7a 60abca1cd8c834bd46f0b6f34e995d2e7eb6c4daae3af2d781986a27d76fe490 dc36a4f0b1c2dd6bc148301b20c84f030b852c8a0f7e7dff1b5f0b26028c50cc a8164d84885a99c6e72baa3034958c79403ea66cf4e8abdf4515a3bf1da8f7f5 5b17c187e623e5504153060330c719ca390af001af5e612c5cba37f798da765d ccc0da17e2475b7ddd8ce0ad801e658edef192ac6b52c766969e7fdb06e2167c d0d4b1775d204104882f5dc0a1381b3b8b418fa84e4e355f105058e13a123214 526c841d15a5ff52314bf7969d883bfdd18578f3178ef398664ccff1bc9219d3 f498eb56c041ad985290dff65838d1fc7a7e0f8385c4545e869b930f3a4be9de 563e0d76fd469c050e123812bb1116ac94f8f5db82ce9e7c4404d80839fe9e2b f56dd43335b4d0673c3d1f597569dd193c9a7bc1b0e633322e737d93c52786fd 25f0a2c4cae985eb96c61e88ed982bb7731aa44950 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 80 /P put dup 82 /R put dup 90 /Z 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 be30812eae30521c3c775adf1959595a43d8a9c4e12d9f9ece27a0c9b5f45e38 8e26a39cb0ad4e25ab8f21b25ad1b106ed22af7947ea8ae4dc17b9facbc5e22e 89a91b046ba4cddde69c1997fa7f4cd949ba3ac4b6e001475daf165b9432dc70 6e30aeead8cc342d94b20a4045a331175655d6aa279627efc0fb53a4ce66b68f a45b5765f0c1368c04a0907c59a0519e67505eba499580eea7d3467b1a026674 04dcba76e4aa7f92f5875ad584698d49dd6eaa7da56ad475fc406f12d7ba5b44 1238e3b5a2477c44012ba0cde43f63a960eb35bb3a3dee5d69223b4834ae0691 6842b9166d8fbfda3f7190e9abf917c5bf15ff8e58f2d81d868b52d435afa0a6 2d812fbd664594c68e7c67996fb1a0b2e8f80877c957676b574074b3b3ffc038 bba19ca3183e79fab9b5a9fdb2c7e1d564fe7b63cdf2c1672c6354bb971bdfae 8373d8d32acbcc740791d51269b4b66dc1f90381085e1cbb02cbc185a2713b1a a08fd3897cc3dc3baa198072d8d6a5c2571f146781321c87a92b8183914d5ba6 8107a205c59803e344bc3ebe618b89347afbb364cff28ec5f4880a71be906a40 349883ec2630df1bc8dc5524e12ebce5cc962f0aae2517b16fa6220766afedb9 a1a94f8d4dfff5cad2650c9ad579f941ad6ef804ca186786c58bc0b6439866fe fcbf3142c9df73a86a58290499914f3b7004b9dc720c0e82891218ef86b10204 a23f34cf5fe80fc29274e8d609cae4dbff5fd0a759cde2db89d3ea9eac00fd4b 54053847a8d82d6232de88aa35c5d8c95563a98f65e6d3544a0218bdb58ef1c5 d6b60bdb7b273bcb81b2f5a191786ae5c710049040f9f2e338f172855e04feb2 20a9532b7f40c9172597938d4ca4dc764a73fd4823b06ff1da546838a7fa740c 31a002da5f99513572223699c8ea8f4f3926177a3d26e34302bfd2a8132f536f 312447e425bad3db545f90576cb3a945fa4234c9ca68a466a41f0ae0e5abc531 2391be5a001b13ecf033bdaa141e40a8a8f953dd35d650a124d144a6e76a7ab0 fbe1708aa7aae02b72d57f1c658f53ae3ff80fced78d7c1381ab459821fef0fc 46b37025c597b5a9bfa95672c0f7f88be6bcc074d4f4321f4a95649140b04a00 99148005641eb62971b448ac0bf87c4df14414fa650fa80bb9fa4d9bfa43b0e1 f742ba7a307890444e7a0948ade22210d623e15806e460049641cb27dd11a808 9c5cbaf33f8a2b8eb5119e9e2b2751657a499c3f48046026163e877a55a5f271 0b56858316b7d4e3db3d384c96369cd81b14b41afdcf91bdfab57577e7cb17b3 32405f75ec5881ced7e230fbe645fbfba2cf4da0bf5492ee2e467f01c5a7eac2 51d63ac1ef90c3b7f5b07bff2da909cc39962d79f420fd0d8236ee4bbc3cc36e 2e3da2ad49e06facee8b648755f5c94d28d9e6b8116ebaa95147c1bd00224676 24e1d584270070f8b7bed8d52e00056ba8ece3935adb21 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 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 121 /y 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 3cc907af3c64e5aaad882c356e673e6ca2509a71920e99549d8797e9aea189f7 fe18bbf718017cd2e32b4652acf12f16e785462980f5895afc917ea83e9c3eab 6297f1de23785f79172538b39984ab261f14ee296a42cca3159e63fb30583af1 911646257a99b4162635fb79e25404242cccc3bf224d4c279e731333107ab2cb 2598dc582dd185669c1fad35d2be0428389a4b175db159508c08a77e842fd400 becb73345663a783f5bd0b5d90f4d2333e92e5ba73a201539fef4d2b88cd3ab9 3ccf9f1d4ea798ab7d1e5d910605cb9df26540071c42c51a1f35bad64e9c59d0 723c875063c27edfcc4aa7ae2b66d0cda03294632e4cf6e9e0799c67267a9f7f 57a7ff2dc592821ec46bdb03506fb25ed7f63a88e8a55bdf14108932b0985dc2 8d160a783beb778cc634481bc156ec4bf4bf74951725196e3ceedaf868f4e71a 7e7be4bc595c5b63d218c622fe5c1a7e7fa4f40bc095c6a2a173f0422d036c16 1b0470e7c4bbd486976ac0835788e88c2443091680c98546bd0f7444c25f3fb0 ba4e9f3d01af346f45d6551199f855e21fdd3d6baf21866e2078875a356ed62e c2a376c3fc7dfb235c029926df03e0c218d6ccf9246449e0473037991e433717 6d00531342c4ff8607c1af83e0d48094b019bd2043223fc82b328a590bc974bb 4cf723f09f7f744ff8071807a6742528a0dcde41c084d7a226b77b815a748532 502e9369c531efb6122ae2aaaa9a6e840be8ad422a2f8bcd3ee14c5bf5d29eb2 8fa6f3da723397e3feb7f97e2746fd0c564892b874dd03436409ed261320c26a 0fff5596ebe67d536493f7eb03fbfc73d0a09980224fcbb91a89d69a558544c6 49067bccf1ba675d6636e4383e99a1169bf02a237280847d06cf3b3a9f1b1450 998ece8fef53c757f3b65186b26c21d2cedc3c8bea3010a2011da8d3742c7345 c0042866e06dce8a47a28cd4fa6ef5186faf717a683f0a5dbdc4679deeed2db5 301334bb3a9bce739b37d214c3a7ad293836eb2fdbec26e0046bde705d1d30a4 74e5a0d156a9121e1c15fdbe08f8084d14814bda89af5e36dc0a87d81a1977b7 52af5e6d818624236573b8b5bf6a063fcd1b129217cccdd7461b4fdb5acd70d4 247a4962201ae587f739545f3aeb735fb187c9a5a3cf8895571d7cf9ce9713bd b1dc977377ae12c5fb6537b1900ecee7de4b63755ecf8490826a83b265a81f3f f190270b6f90eba05c58a89fa9d74ffc84eb66dfdfd9a0cd5b5a50a8e0a069d2 d35ba108cc07562badc01ecfcdd5a7babcf9b553b440974fa2fbf0e6ab4cb5e7 d2d7c94a74811fc2bf67ce2edcc6af3fa79d1c8f44dd4ad65ebb82eaa40c4998 0b5e6fd0ee4b72ad78023bddbbd35193f0b1c32d1016adde38da57fb5bdd0a6e d14e025ff4322ab84af92807ca019c7a6f8c0b2b65aaf5a09b359838137961e1 cbebd34eb73c5656f2428e15f38c4dfec333bc95b3934c76b7baa12a61be941f a2f36016c74eb6f76a98c22003cc8b7e7d0bce14a1d11afb18f128fc713383ec 20d1cd8ca7d85db39e99c475f10a0fc13d8a40e3b27d73b632c7b533696e163e 155c9e07e47e3463d410c51719a43b219251d8e60f61abf5432ba36fdabbafb4 c6592a31065659c0dbfc9229a440fdf7abad5b3ace24d311255d49fe45cc9186 e5e94dd0605b5bdfae111536d5dba60366f09e1de69f25d7b323c3e21c365e28 c0b3d05bcff8febaa99216c3ab404d6c8771f3eb2da66a0160eebe654bd31c29 f6d089e1d646405a9e481076c2951eb7f27ba675d31160c2da7d8ff10ec62cd6 55c3d0c738e2c731a84957dc00d22c77b98a3ff582acf78e82c152a7a829a6be 728c2dbe7d093ae3de1db316b5b4311455a95ccf18f9e3af9525f9197f5b294a e744ae2a82662cbe691c711b25ef1bd483bde9bca4cbcf8587edd51e47999103 33da0fa4e4a5e4cd46c08a0f44652c3e07d6b584e5e32435c46dec082476edc4 c80e75024d766177816f73beedc9b39ee22a86c07e8d903a21698266588692e4 8a40de975a761d6295c369aba2365413b446ba72f7027216d4eaf726e3cea101 fa9377cbf3069c9274d75846125fc31eac4f1f47d7f1a6037fff9c98115a089b 6098d5eb1b5ca8741da88f338c110ff006535aacd7c42ca04abea64139eb9ef3 ed903dd4c42ffef36db882c590f9b4e6e8f98cc7819992e868a4f27bb3e7a3e6 92982b58d2cd76e7f9aba863cce3a55b84f54522a7d28a978c9fb329c0f6eb3d e03077367af2eb07a29cb78b6281fd711878db7320e85587f8a61d5682ee0029 40b0512e77085ec5930df0e1bd4579627d1a169458eb641ee132ca77f567d887 3886bb1fd26d48a107bbf2ec1b3aca4009bda0624b26ec72a36acc8b85a518ca 68a3a0a50c285550364a49ef4d479882217982e07630bb21189b888313130026 a29bac7e1ac00bdd0c6155cd2a1418b0d9bbe24dc4817d01c89fe2d081a84a30 18999c626439aa1cf5a35416f0957552b7b2ccb0c454ee655c40df5f5b64363f c40d675eaaa9e67d1b14e38a6313906ab61147c9c8a06ddc2436bd411f9310a9 3db1244473d41c5094e35a3ce573565156b28eb0644892f13930db1c5ee2c844 a0e11c19d2e57711d0b75e8e5a45c4d853735455d6872d808b53a1952eee4f73 8a02e4beac08c858019da83ed0f81ba08c453a7bcc580eae419b084818dfa792 55ba8b2247ea138d2514fb5d9e26bdf2e389b2cfe038b0b5f23120f15a28c13a f24ef7cf18c0421ace91a2112ebbeacf14561a1cb49a044d001aaf173aa62c26 4d47467c6124d92bcca5ca124414c840ca3cf0d28b872a1046aaf52390d9c9a4 3c68c18e8866336b48a393d6dd2b41e5b17ae1e3d58270890562a91f56e4ce4f 9e0163da6b6e13fd4f526084de20d2e011b11cb38f789333490d2959d92b9b26 e26c6e88c615cee50a5b274dec38925c73cad0eab909bfbb8019bbada3739b58 eb4d02711c94774f01bde8c4ff3f1e9b5461ffb8a0f88f2874218d161b9eafb3 4e0167c5d93708a59ca4d3130a1e622b031246dbb6f54996c6dbab1924a07102 88fb276618cc59d9a7383c9c5458c30412ffb83c8074326266916ae575f8623a 1a1bb8240a5e82676dadbdeab3863d6c9101d14df9583451415b5374eedcf79f a445139ae933707f2325cad7fb5453463ebeaeedaf2084115fb848c75ab4f9a6 22344e6d000c517535c6936ee4aa0d8ab6eb325a8951c8fc98f2207d63c7c4d9 84ea5f62992d4d8ffb9ac6a4bc39a0f256ae0f50f9098861d003c44e6add03b6 6dc1b58059c45f1a292099a7156ad9a0bb79501094b41da00ab76aab42893a51 13e5e579608f3ad9480c31a0d4ad8d10c2d8e075b1d22f3c9cce56846f6f864e d265a96c7bd3f4f4f9b982ca45975f2eda78e10455d38586776c155a084f6d9a 78c64e5d9d143eafe0e44c4afd47ee33c4eb71dc8b0a017f7a74006b2a6b6422 fc3bff242eb852e21e73627b6413064526ea21018fdcf246d32122849f6fe636 762ffaa41d0720f379a854efaa1ae2769b8a43f3e9f3e21a43ff26ee0b64e896 9c617323043a69527fb4ca7a63a963efa98be24f47d12d144be67e5cb08c1347 ea993a33bae03e1455cd4411d3cfe1b66df5e7796c4086dfa08d7f03997a3b01 4f915c26281cf0647d9d49c65c27916a715785f48e90b050c6eb06b2ad3b0ce1 458ca89d1bbf4bdcfd16ace6469ef2d8438be0e30cc0d30739b41d3ad34dd1e7 eea9f0a4f4e1cc88532289fb85de03f6dbdc5ab76df6af17467bae325410d512 6d49b262328e75ff9a46731b47cbb4f707f8bfeb293cab8e376ca968c6cd84b8 6cb7e33825be2b8a7b613aab8515f3d0b7f9f9d842cb5aed6a8bce32bb6e3ba9 df5cd33da1a03d473319f9822adb84f8b5cb133e75bc009e00822f901b680bab bf91d5c425e4021ec53227cc5fb269f2c52fc1169e13fc9345f7f008acf3be63 c3154610bd8990e76a9b938928b1b4348ece799af1bf563e08d11cb8bd6af60b 6c2c07b6cc648afa020f764e94837bd334d2a516e5665f6c66dcee2ce3d30c51 86444f1da803ce7cc679b02fc9691afe9d9fe46be30dfdc659fe29fc97ad2f22 feaf030c30dd207104d856d2a7c7a410a022fec305b284f901a3b99367e09233 7df8f14f95b0aeed6cf11e8b6dd3be027198a4e647572e48852fa28da4ced700 f327f4c9ebc095d0e0c15d39572da3ec1ef8829f36188367a80547091b46ab34 038dba769b5c7636c442c273c05ac6c9ce59cd9675d0fb1be60a23fddac6d035 6b68696029b04d1429e9f62fac51e5e27a4c29e8c2ed270e25ff1eb4174def72 45e381609b7dcf12201ece2a5d4f1cacb3c9725542c4c188d097534f213bbe3f 9e1fb7242fa26e848a8212d33efd2fda1d3e2b7e64feacbfca5fe5d597073603 9c9e4f0a52cc619f9a4cb81caa9dc530a62ce5e20a96b8cee543e3e6a42a0543 4ed36b770246651c438dd6c7e028e59189d7bdecd5be2bcf3a144e388b452382 82de4895c534b5ba50adef70746394491d521a7e81b4380a708a4849f5169159 f27ad68f9f46f3ba2ad2cd9b9a39f7b7e0470976aadf87c1093de11adf7060d2 c5d58b65977f25604369d123e02595374fb4bafec4a44350b1de7789b902e507 82020346e0f7f586ca41d6e93eb995bf81f9e522afe30e49cd77a82cfe33eed1 5179de20f70f77def71d4420274bfa78f3caa70b59c9adf7d0c20905283577df 784716fce72d616824da9ac519e20fa17a7bc54c6139cc3d1e5755159731b988 c86cec6a6cdf44ad83dc169c51e0bfdc09ec3a227d9e 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 48 /prime put dup 49 /infinity 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 b910bf03a0fd0ff582d7bf32eef7d0676dea7d3546c46d4eb62cc5f957c41086 7b839cc408660dec5f886c9f9a58ede1634780b7da0d8f10935518e756f1f21f 1e0e2534abe0ef6e0b849566e60b6487cf6710f0a76512c9378cb10c705a88ff 10bf1b4329bd9bb5978ececb15fde07c89c703f48e016526d84f14d559ceeed9 7180b3d624f013f79c7512e73af9846a8e6d2c5fe7b97b39e21af802250e4bc3 5a31451b09fbb3627a3a48113d2fd144d8e194cb7fff0554bc4adf47a6acf5a4 b78d0ebd72b08e83730786035a6101fcfd8f5c0afd20e79a1a27985cc70e535a 873f1c609ed2a4769e12a888740ddf5d65fec70b58a843ed3bb5ea18b1ded7f9 50caf2de411b3d6994ae6a4fbedc98c472b4e9387da5abc0d4c4ade478197116 e34bc2fc3f8b09f88f881a5e1efa73ae2039f93ef4df11f441ea33f199ba4992 703c9c88e1b82f4775053befeccfa322a657c4e68ac44071a0aae60f818b38fe 8f5fffbbce23845368bf485c19e8afd52f2c8907bf6b7ace72b238c8fed0cbf3 969333ea56956c1222e0f6b6331cb3cd7b6e986ddb91c3af6539b7b6f9abd789 69603f54a30efe30402e15138ae9a7543effbb189d2e17a677f7c4a61f33af7c e4979988d0e77d4c459a731b0b09cf7147101a51cac9e226ad6e28d2285855cf 6adb9211fc8bbb2df0c821c78707da1dfe0869827420d9a577205332727e4bdc bc04be8694ecacaaf3eef489a688809c48b8a851c4f025a6dc947ce05f710202 f09cc87923cf16e47d14288ef3b75208ab4978077cb79bdbf7361ac42fffa3f5 f28277607daa29917cf8e54d9b9a722867a8073867c37520b4cf40e3ef88ad3d fd06f7d9f8b40c9ce8d8ee7d46652e5ee82e4888f2df95879e2d73fba4e31f06 b8d0e6bcbc3373d48ee51c989616a4196f3ec2fc0f421371f742b2f8ca8571ae c49cb639bc2b7310cd5f043941d5b2ced680e447c49d32382892564bc72a 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 20 /lessequal put dup 24 /similar put dup 25 /approxequal put dup 26 /propersubset put dup 32 /arrowleft put dup 33 /arrowright put dup 50 /element put dup 54 /negationslash put dup 55 /mapsto put dup 65 /A put dup 70 /F put dup 74 /J put dup 83 /S put dup 88 /X put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 110 /backslash put dup 114 /nabla put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueXX 5000820 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f67052788e6703bb5ee49533870bca1f113ad8 3750d597b842d8d96c423ba1273ddd32f3a54a912a443fcd44f7c3a6fe3956b0 aa1e784aaec6fce08dae0c76da9d0a3eba57b98a6233d9e9f0c3f00fcc6b2c6a 9ba23af389e6dfff4efec3de05d6276c6be417703ce508377f25960ef4ed83b4 9b01b873f3a639ce00f356229b6477a081933fef3bb80e2b9dffa7f75567b1fa 4d739b772f8d674e567534c6c5bbf1cf615372be20b18472f7aa58be8c216dbd df81cc0a86b6d8318ca68fe22c8af13b54d7576fe4ca5a7af9005ea5cc4edb79 c0ab668e4fec4b7f5a9eb5f0e4c088cd818ecc4feb4b40ec8bd2981bf2336074 b64c4301d00e489955b2ce2d9040bb11626f0633ded65ae053753f4af570b6da 256678ac3c38dbf0c3558b68ffd0eed303e91d8e6f146bfc1423c8e355d8d909 4db424413d8433175726ae6fb7d5792226a4d8896d0f0fd5c10d1ce906a2810f c0d86042ca52a9ed79c72f1680284d6a7539d80f52dc23c0d2359fcf778b040c 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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 12 /beta put dup 14 /delta put dup 17 /eta put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 26 /rho put dup 28 /tau put dup 30 /phi put dup 31 /chi put dup 33 /omega put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 67 /C put dup 68 /D put dup 70 /F put dup 73 /I put dup 74 /J put dup 75 /K put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 83 /S put dup 84 /T put dup 85 /U put dup 88 /X 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 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 115 /s put dup 116 /t put dup 117 /u put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-30 -250 1026 750}readonly def /UniqueXX 5087386 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 dad7b4e6d6f9cf0688b69ba21193bf1495807e7bcb09b7064e91fa0ded228e42 09aae407a7aaca60b1076299ac4abd23ef02f108765f0e3d91f92f3afbfded37 2fcf6e4b1416901517da8f2fb3c9fe7a87bcbe6fd36cd5b5823fdb74229036a6 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSL10 %!PS-AdobeFont-1.1: CMSL10 1.0 %%CreationDate: 1991 Aug 20 16:40:20 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSL10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -9.46 def /isFixedPitch false def end readonly def /FontName /CMSL10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 48 /zero put dup 49 /one put dup 50 /two put dup 54 /six put dup 69 /E put dup 77 /M put dup 80 /P put dup 97 /a put dup 98 /b put dup 100 /d put dup 101 /e put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 121 /y put readonly def /FontBBox{-62 -250 1123 750}readonly def /UniqueXX 5000798 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e 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%%EndFont %%BeginFont: CMTT10 %!PS-AdobeFont-1.1: CMTT10 1.00B %%CreationDate: 1992 Apr 26 10:42:42 % 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 (CMTT10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch true def end readonly def /FontName /CMTT10 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 64 /at put dup 97 /a put dup 98 /b 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 readonly def /FontBBox{-4 -235 731 800}readonly def /UniqueXX 5000832 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 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ff90c0d320f61ca0e477e37dd2cae5d03f97843b18ae2d817a29cd22431cecea 48f63b587df6f78cd10390a90b7e7612d0e1c5cedd4c9c98a8f81907bee02eec 5080e2c7384c762d0afa4adc86cdb4c5b6a4 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 0 /Gamma put dup 11 /ff put dup 12 /fi put dup 14 /ffi put dup 19 /acute put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 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 87 /W put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put readonly def /FontBBox{-34 -251 988 750}readonly def /UniqueXX 5000794 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 39 /quoteright put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 71 /G put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 88 /X put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 123 /endash put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueXX 5000769 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be79011d1f5bfae5c1f476ee6f05eb1d2caeb 269958b194521197b312fcced4867f3c8fbd030bd715d8ffda1dcd454b174e7a 1a97b59fe770e67702519d9d9b23d61ac08424d555242a8ca08c49aef300945d 99b999a79ce74804ae6bfde623f4463371442f6523a5f6ce19c839a708c02513 2e22c696c8ccade45680e5197189d0f98e7f0d5f955e353970b392cf530a68cc 56b0035ddfbf206c3074beeb0739dcbca272a6e629fb7aea2c5ba7bae50c7b4c a595df78200c352997ec3ee564df229fbb5473f5e8ccb1cc0153e9a7e299a8ea 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6a2d65e9f67004dedcea3b025f2a4df5733dcdca4f47fedce359c4751a1f2142 6272a8b822904c171b4afadf0da0c0e7baf465b3875ba39c36afa11fad7edd3e 03399b7fd9eac7ba55519356c5cf8028b32806c47859dbfcdf676b2baf584e2e 92c1ae250f8a0c 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 34 /quotedblright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 63 /question put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 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 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{-251 -250 1009 969}readonly def /UniqueXX 5000793 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 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44d2e51b48b4ac5c52096c74ed369fe5cbd58357543457edb06c550102be74c5 b38f74abdbd8a889dd54f44a3e8d1723eb3336601ae3ee3b820ba74ce2525069 7ba355babd410dc821a554ef848b3618e016fcfde8c00375461508a40a53d186 f841b16f5b42254570569a3de8c2795ebfa0bffdff1d3bf063 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 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 65 /A put dup 97 /a put dup 98 /b put dup 99 /c put dup 114 /r put dup 115 /s put dup 116 /t put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueXX 5000768 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306462c563d51ece66fcc9c831843ecae 1fefc1a232e2724f7baee428ae03aadb95c3035345c15e9922fe49e1f2cfc980 237316572dbc57064edac9b0db8913a5e2d45e97e19a91435ccf8adfc835b585 48e74b291446d689c7f2f8c4325e8356e974ae30c3e2977477baeaa33d141fb7 80f59351e84bfc88c87b3a4a1d25e168b9d134554f6a581378c7d2d6eca8ac09 045cc3a5e0ed86f147133f094029e3483bebc81cfde69942cf645ea20d0a2b64 5140ccc2f9cb4a3d969f4b0a79cd3329b440dfcff5285494406f3dd08d804c86 df191e57232f6f1f83df6430f214e03fa5412b1e13d83c60c52a698bc9a38159 19a64162af7b83a9468fae9507c6cb55ea80df8c696dfbc3296c663902d77e89 a181502627281181ac8624f65be9de5fb32f056cc7dd7cbc7366a91717b5892a 34ef6b441871c6993da3a81afc9bf70ff3842a21a29c0e49da1ccba37dd6b920 2702db510b8b1109acf4c27ee8d676bdbd16abe189f34a9b61532eb777457698 cee0561838eaab2d3939c5093afbb215bac433d1918c739b62566906bb2187f3 378521fb52e70c18886e1aa15f53a1f8a169528cf2ddd54c203613ff88e619f2 6c9c50c2c3b8b8c868a88afab1dc8472e7b6f3f992a161723a3c0904865a7c42 3281601eee91f35222c4a71f343739f102eb43145a47e4fac3bd666e4c114a90 5950618ecb1bdc29f90f6ebed1d8325fe6b52e14a8870e9b4bb088e585c71b91 493c12845a101937e3473ac5e9ae24f7c358f06c8b42dee8ef2a1a7a9f536037 57517f2fb6da0ea305d9d20cc74d922ec4b95b8b299459b71825f6f67abee3a6 f5267c6283c7c62aa8d36663071f18e1a3f0b35dcd88d739caf8dbc344630a28 7f583f037768d861f37d7929f248146b5f31ed9f61ef0a9659b50f47ec328516 9fd999b19687e50303557945476fe9845e5c4780045d84d7df253fe61bf69607 fd277a2651e58aec586f836902e8a7d11babd27ed000f14269d6b8774a8e955d 0333ad4d5e8eb3a3c13b2f8be09bd3d3b0c90524bd356ea9d807fcc33e248bc9 ef272d6eb1720212a961ed6ed6b5608cdc2a877539cc04c7465024fa19676193 0d477363659fd7f83cd9df1e11478dd6986b938726f940538e005b155106ff49 3ff4bd4d71f3347cac3efefff4f7754c47c230155348362c04864d2d6dede000 6a1c1fb0b7af9b765f3596bf360baca480cd8fa2a5ef590e6cc5b12f0902609c 5eb42b4127c43c08d3983d60b79b1949c7240250519285fb98b38261097b8b27 7b8dd67b4726ca3bf8c71c37ca5fe5ea59b5c33437d9693998a230cffae1717f 87444c5d076a3da37b1ffd9aac09e00a52f2373c83b325639629fcabd7da7098 135addd938058d3807b6ae4140440420947b6a75d294009f557317e1214b6cc4 b15d656d52a458ee0d560f58c14b805760c9f077e95e7d1184c1658cd3a7ecf4 9d9bc517952befbfb2a3e39c8d176ed55e471eed8e63ac85394098109ea7b165 f2d28f15305ada6a921b2431b33c663efd894ca057dea21be2b1467e367cd942 6f416ceff3e6b0a26fa7d3cefb1240e4b12fc7cafd2d0c9252b512537aac6efd e5577ac553f209c64a71af0c65f63912008a225fa10760ba166c9ce1d3d79cb0 f17e3928e57a9c416f47e19bf6b24e06c114ebfe447249385d99d1fe7ee2c852 159490d28fac7c6d1044aa7e1e2db944764596fa918e0a99e4c84b339a1c7565 f213578a7aad7b7a519225770e06da6ebd2177e735bec8358cb26294b6294f08 f325c62482bdfff3e2b0362edbe74097a46b3bc9307deee656b003a5ed3f67a1 770811f5c1594a763fb88453683f272735abb3623845d2ce2b8bcb90fdaa0d21 8369ba3a3bf3141b6f9d7c95496e6fea3cd32269f8300e8b8ad06c89516c16f5 8fb909e5b77aa630a6d592affad13436b41106ff491064d36d9f11bfc0017454 1cf515421d6a5cc4ff80beec81067ace70a2503dbe033b97f72bc80c678df348 2e68714f1b0a112559d5b52b6f1c5c125ef828a1703f23a5ce238ea668a3ce05 2975ac3846d3cb05044ee131fe69b65036919eebcd10e23cdb64ff27b7d96027 06c79641156c92eaa0c222df8f27e090815fb6cb22e317cb866621dd3a9efad9 8552a0a9feceb2319dac7f8462e322783eb09bb554c4f5bb343554850a3513c4 5e1117daac11f86c32109e98bfa43006c55bdbb11d60c41a686b9db67b5f1556 88bc2850580c7903286b5c8a6aea012d79795aadf216f4b7bd218b7169407d74 e0ebfd4a75e084e8b56868d5e51dda029626068b6545ad5e09fa01c93255ef4d 3ce811d6e4f5f5305aca4d2ad04d966e3c93dbdac94c6dc261c38f2fe7b55166 1158fdab0df31e3f260e0e8548ee67bd395ac6acefb65f7ac37c0c63126d554e ff916ee505d6c4a764172919c8bd101df51a51de5c14ad35d29e6e04dc5905bb 695ae1db14a0de5f22d6cf3efdeea628a4be5a1e45d318be6b4ca19b9fb5ab0b df3c72061e17fda2a5041add08b6044b2443bcc3b2d0540277ef4ff3010156ef a85ee4d513c4c523881d139869dd282fc8545d5bf8a6ab5dd12088f5a6759328 ba3b9d7fa63f08de6ab6f944cc079106e6795a42ae2329a181883c2135de2b31 4b32832a85315c19857a8b30a25f4d946155067e1a5952a6cb1ae9fd8a3922d8 730a496b33f8f01392f3483c5792d2e284a4f531f4102ade5a1bc26d8d9d6c86 bce19a7610718cd776bfbe288196c8d5d3c0f35ac0e8a67375612783b72b2988 b97c05b7431fe1c5659281e9e1fe1840a79a548b5aef9d8939e224894530bfe0 d101dffcad4c71726d5d9cf628ad739b79b95bf2cccce9db26a1172b7b3e1561 0ed8abda208dfca64db00becb320a02afd5180f72d0801befd3926e5a9ff334d 485b5acf8418facb9878310cc5138b92552a8e5cbb5062212bb0f4fe80ab5c9d 6942dd7bcd8778d0f1d29ee897535118848a3cff1f87a7bec8e8b4f0fbc460bc 27789aa597ce8623039d1b798f72fab189ec6ce5b318fefb2072f5653b835102 c8ff9c5d6216e64e75699126e04b5cd54f2496e2641535ae9cd92df432151375 1d1277db1f31d5d5a87c35f18c75bb61492c512d1c49907ea90b265aeb917df1 6b9346d1671a2add7828091d0c60edb471046d0359ccdb46f5b234cc97ee7f09 5465b73931ff3adeb1c05fed55f0ecbe00e0df03b8e330d31c3cecbaad47168c a26ef9f1de36af7875059b2ad19367e9831952a6cffaa1603f5223401e61483f b30297902eefd415e3b1388114b91b91df35684d4a92af882a4641a17deb3543 0f5f505181da99edbeb02e876d1f63352c753b3069db41f6633304ede621d153 b01d5a1b7b9df44d0aea849209609aeb2d086c06ee25b24347e4bff813f89f85 ff19cfef7395fdbd760a35edf7e25fa95a7df39287bf43bde637deafc2793a9b eb966c556654d721abd2cc3c93ed0c415fd3c9810cb1276f2ef1f5a9d2e92c58 88ae1f9a838657a4ef5f8c3e38508b3a0d345fd431688d09a3d6c7041a592bb6 45d06695c6b973b28358729604e72ea8768944cebe4ec1c60601be55b13cb9b0 bfb5c8f3ce2ebeb4142c9c182a2f9b483ededda2b0267e24dee7664735b9405b 3c5c8f6eda178131b57bf46a3c14f5877ac251561a782d4a97ed2bd1ed2c38d7 36a7ac98ac17ddb03fa6077b05e1bafd4983a9378b72f90e3eea4c960fc0d69f a8e25659c454dc66d16805ce047448cd57fea403ec73c2719f1b878911e5a664 1bed6dd554c8075abb51daa26bd86ec2e0ce54e72b23d6b1a968d97f46f4b656 16c2b868bd24d32508508018dd4f2f53c875d1028980d3d25f09a7aea0cae746 f75c00cb4a4a9cdc94a064b6dff0366be598f33ce10d05690b5fc53615edc084 641c5046f2286bf35207a643fdbcd0ee996cd98817280af76d53053f5a160f0a 0e087e5ea3981017bd62601fc4c1b567108e05a54a857817f282ce467d9ce422 16ed2d8b6b369676ef3895a548961e4aa322a66b76a768f7368da0f1e8cdeace 75fc480f0e649d5a4605bec448b795695611b0e181fea50265ed04a07e12aaff d466bc234c1f6ead40298bc99047e82a303ca949c6219e003881cbd48de389cf 7ec5d30d6ab1bf91948afbb2506761faf5e9000eceacc33ca09e49ce0dc36058 d9658555e497b124f69f3462edee2522acbcb76d8e2170519fb7e8bb8d6611f7 da25ac7ba483145983e7ee4c3f191a2f57ad95406a4936ef59e72354fd4b668c 1071f6d87d1a68890c509d7bb77d01c9ef5a37443745112709a3bb73a47e303e 0e5014bc604ce71893aa99e58ec860f87049dacd28b7a52ed59ab033fd8471f6 24a027e46c15e0bfbc2211375915e6f36b38a901e319acf8c76b8c239305f17a aedc8af297264367265389afe4728806f515c7089ed8e52fec072f4866c00336 f940c9db719e7154231bc2b1d6ddbed27d25f1760854de8b2cbd8a7743ee0180 fc69142608bb0a39560ddc8bbef8101db97568beaa4a0c5f323fd32ad7f2f5aa 391e6855e8d5d1a7253a2eee3349b5449a7c0c7cbfe98b321fd93d5f733b82ff 1c719cd0b22fe9e217fb1372fea8b9aecc9e2178a58e1b90276de1bfd8a9dd37 8ec4b80c6da43d22feb2e3786bcac11479c1fb83a5dd7079f89c2c51f06b16ce 9f6ffd619a146e38ecf8240207408c120aad6fc3ed881d6952503201c4a85a08 ade153c25d16a080b5a6a8fb20058f496ef4673b978d548100289934b278cdbd 647749f57466aad4443608ca8adab04ac5e1e480de3925fc97c61b942785e42b d86b9b51523335d85508e8632ac2dad0d1c5ff7c99c5a4e85b16b4e2494f8a78 169722b7558de48dc239a670f84cffe7f70e40519638e85f5c6c2a1dd229fdca f239532dec311fb16269101c8ffb8a7d20c22ef1ced8b70de4370fc7bfd0990f eadc96398240df157482c7e2b4aa558020980be9cbc979c3087b54b235e40af1 1d730b3c37cff72f6cf88ba1a6aaa23ee2f894c5e5bcf399855cdca397a18942 ef8025410a12e486ed9cbac2b989106d7217fd90ac9609e88cb8cb6ffd2811b7 c03d4bf139a6539a4993bc0591608c41fcfb9661334720115ba292c1c2cdab66 d59f2baf6e739e3d4904a22cd9001b4a3d0926ba9d67d4605e40aeb8947a421e f27499b2b4172ef8124e5a6655db8b9874b5a0cd3a80c0a4ba9d011e68adb4ea 7393d94f616cbfeb78b35fa0d73add22b7010840dbba38ef867a1adfa3127542 3b427febfb2dfc80f54e23a03b3219a62be337fc96 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight 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 71 /G put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 88 /X 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 122 /z put readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueXX 5000828 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3ffd20326947b1065649b533eb5e9e3 a88a87fe987918a687950b26f647d6e08bf14d983ed1d4a8ca1e2da8d985f944 2eb916f5b6ed0697ac7c33e1e36a3bf460d34ce45f1631871097cb04f18e3889 4cf4ac1538eb19481311d24fe3be7beaa4a3730e8b4831fe59d6d9ce2e46116b 629c7ba2f9ce3ecba2f43bc162a5a077ca1b2882a42afdcec3f4b75b5d63e0bc 8e5dc95257766d8ea467ad9cbaf47be60f797580cced6884b3a68f70c91f4fdf 80fd00ed9139e7f480dc3a76af72ad9b434187730bdfaefe4cbfe5c7edcaaf24 9204fd703011932e5a3c27be468b7dab69daa18dbbb6335ab8ddfdc607961c7c 02582763fa069d43563ba17704029945cf42fcb19cf78c51df0ec4d851086d43 c1ab38e865ef36865fff3a08b01cbcb070ba4893f7482dc7819ad03d337e520e 7d8cd83bb8ac7aff4df36751f4f12aaaea2b6d7260c09a26e987c904003e0723 7dbcc4a8e4e85cd3259f40d3caf55c742cbb40ad0db1044c20a4f122a63f7b81 1d945bfa69662189ce4d089464d2ead9f6ed60b581b04114d5d45e97be27a576 35154e9b8464f2a0df18da855eb51b8f64d3054b0b2e84621f025a0776d16fed b988a43d2d64c32768dd106b9f541ae8e68ac0bb7399e16f4d4ba074b2152d0a 4e9b5bfc21bfa0d531e28165e518ffc054dd4ed0d74eca079d20e79670419f92 e03bcbfeecf3554d661596f6f29e12170804d15a250d917ffe7a8b6e8fa53047 1ed97ba56481589a65ca13619129f1e26affe6c22f0c38540f8ce37efb6e9e9b 60b51a5a9cb4454fd7c7f92f2f9437159687ab590f35b946a54491a7739c9269 e0f1fb48b129811c82edb3171cbd5105fe65906974db8a7da92f4f43dc7f6b36 6981146d86d23613a0b38d9d6191db9fea0172e982cb1cc50e382f69da1b377c 1be9407f88a2a761b56f398467a057f35d18c01cc377e006182c891ec80272e2 2e3a5c7123cd27eccdc87392a9fdf63bb40f02b7649cfc53ef933c6518744cb8 9cbbd437830ef5b1d58b00b67bbcf4426dc1e6ef583b0b4ad3aebd741e3ff0f7 29ae271f032019b1e7268043fd215be590f9cf24da14a90f1683afcba8006cef 5d5632c2481fbc1bd8224e8952debb7c4596a237e5505ac143180956ca1f603e ec70cdb4a33e1c4ca174436a38b06fd9d2b9196f02114b77a9a5f66595727faa ba323668573ae328f1f84e5299e9eebaedb380024f83e0ba39519bdaffd36eba d88c26ffd8ad02b0f3b11cece842d5a02973232999932846d95b6eb507bb39c3 d2bfe42904065c637c3f27e8bfd224a50d1d6adb4fe4addc41cba8253c3ae97e fa1dc8abb62ba4ab86b7a4ff9a810fb0ecbc71c06e9089fc42a864d6f5aad7e2 276183beedef909cd4a4fa2c7268cd8725bda0bf1895a671dff05e8a76de8004 cce04fd871006ccfebd0e08e0541af85c96a5db4018c92b440a66ea0c4d8d0fc a5271ff144c6a43d2f334f7c0f9ea0fda591438ba271a63dad588cd801bf73ad 97b87815345f3b60dc4701dd38eeae3c214117eb03b82f1d51fb326f67ca490c fcc6dec7699beb08e736ad121a5e2aec58aee88968ac9b2152ab5961f8c87189 d0e80435f85e5480af17346106555c7c7ab822a309983872c5e7b8d8aa640c29 3e644d596d04f9d0a6a20a283997a8829697e21b7b49fe4431ec76c50d17cc14 22db07bb32c6ea14fe07475784b070caf9b7681cbdbd47aea8e71d54186d9311 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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Fl(\000)g(j)p Fm(x)p Fl(j)2750 1333 y Fu(2)2789 1369 y Fm(a)p 2244 1413 815 4 v 2244 1505 a Fp(1)e Fl(\000)i Fp(2)p Fm(a)f Fl(\001)g Fm(x)h Fp(+)f Fl(j)p Fm(a)p Fl(j)2869 1476 y Fu(2)2908 1505 y Fl(j)p Fm(x)p Fl(j)3019 1476 y Fu(2)3100 1436 y Fm(:)202 b Fp(\(1\))166 1829 y(The)33 b(presence)i(of)d(the)h(nonlinear)g(transformations)g (\(1\))f(mak)m(es)i(the)f(problem)g(of)f(con-)166 1949 y(structing)e(conformally)g(in)m(v)-5 b(arian)m(t)29 b(\014elds)i(and)e(esp)s(ecially)i(conformally)e(in)m(v)-5 b(arian)m(t)30 b(dif-)166 2070 y(feren)m(tial)j(op)s(erators)g(v)m(ery) h(complicated.)166 2295 y(In)41 b(the)f(presen)m(t)i(pap)s(er,)e(w)m(e) h(study)h(the)e(problem)h(of)f(constructing)h(conformally)g(in-)166 2416 y(v)-5 b(arian)m(t)38 b(\014elds)h(and)f(di\013eren)m(tial)h(op)s (erators)e(in)h(Mink)m(o)m(wski)j(space)e(as)f(w)m(ell)h(as)f(gauge)166 2536 y(transformations)30 b(that)g(preserv)m(e)i(this)f(in)m(v)-5 b(ariance.)31 b(W)-8 b(e)30 b(use)h(the)f(geometric)h(construc-)166 2656 y(tion)g(of)f(Dirac)g([4])g(in)h(whic)m(h)h(the)f(crucial)h(fact)e (is)h(that)g Fm(C)2303 2671 y Fu(0)2342 2656 y Fp(\(1)p Fm(;)17 b Fp(3\))29 b(is)j(lo)s(cally)f(isomorphic)166 2777 y(to)k Fm(O)363 2792 y Fu(0)402 2777 y Fp(\(2)p Fm(;)17 b Fp(4\))35 b({)h(the)g(connected)h(comp)s(onen)m(t)g(of)e(the) i(linear)f(orthogonal)f(group)g(in)h(the)166 2897 y(space)24 b Fi(R)489 2861 y Fu(6)550 2897 y Fp(endo)m(w)m(ed)h(with)e(a)f(\()p Fl(\000)q Fp(+)q(+)q(+)q(+)p Fl(\000)p Fp(\))i(metric.)g(In)f(the)f (Dirac's)h(construction,)h(eac)m(h)166 3018 y(p)s(oin)m(t)36 b(in)f Fm(M)46 b Fp(is)36 b(iden)m(ti\014ed)i(with)e(an)f(isotropic)h (straigh)m(t)g(line)g(in)g Fi(R)2712 2981 y Fu(6)2751 3018 y Fp(,)g(i.e.,)g(Mink)m(o)m(wski)166 3138 y(space)24 b(\(more)g(precisely)-8 b(,)25 b(its)f(conformal)f(compacti\014cation)p 2390 3060 105 4 v 24 w Fm(M)34 b Fp({)23 b(see)i(Sec.)f(2\))f(is)h (realized)166 3258 y(as)37 b(the)g(pro)5 b(jected)37 b(ligh)m(t)g(cone)g Fm(Q)p Fi(P)1489 3222 y Fu(5)1563 3258 y Fl(\032)e Fi(R)1747 3222 y Fu(6)1786 3258 y Fp(.)h(Because)i(of) e(this)h(em)m(b)s(edding)h(of)e Fm(M)47 b Fp(in)m(to)166 3379 y Fi(R)238 3342 y Fu(6)277 3379 y Fp(,)35 b(the)f(action)g(of)g Fm(C)985 3394 y Fu(0)1024 3379 y Fp(\(1)p Fm(;)17 b Fp(3\))33 b(on)i Fm(M)44 b Fp(corresp)s(onds)36 b(exactly)f(to)f(the)h(action)f (of)g Fm(O)3198 3394 y Fu(0)3237 3379 y Fp(\(2)p Fm(;)17 b Fp(4\))166 3499 y(on)39 b(the)h(isotropic)g(straigh)m(t)g(lines)g(in) g Fi(R)1676 3463 y Fu(6)1715 3499 y Fp(.)f(This)i(is)f(wh)m(y)-8 b(,)40 b(it)g(is)f(tempting)h(to)f(consider)166 3619 y(the)30 b(relation)f(b)s(et)m(w)m(een)i(the)e(manifestly)i Fm(O)1766 3634 y Fu(0)1805 3619 y Fp(\(2)p Fm(;)17 b Fp(4\))o(-in)m(v)-5 b(arian)m(t)29 b(di\013eren)m(tial)i(op)s(erators)d (in)166 3740 y Fi(R)238 3704 y Fu(6)310 3740 y Fp(and)33 b(the)g Fm(C)738 3755 y Fu(0)777 3740 y Fp(\(1)p Fm(;)17 b Fp(3\))o(-in)m(v)-5 b(arian)m(t)33 b(di\013eren)m(tial)g(op)s (erators)g(in)g Fm(M)10 b Fp(.)166 3965 y(This)37 b(relation,)e(ho)m(w) m(ev)m(er,)j(is)e(not)g(straigh)m(tforw)m(ard.)g(The)g(Dirac's)g (construction)h(con-)166 4086 y(sists)32 b(of)e(t)m(w)m(o)i(steps,)g (namely)-8 b(,)31 b(pro)5 b(jecting)32 b Fi(R)1828 4050 y Fu(6)1898 4086 y Fp(on)m(to)e Fi(RP)2248 4050 y Fu(5)2288 4086 y Fp(,)g(follo)m(w)m(ed)i(b)m(y)g(the)f(restriction)166 4206 y(of)j Fi(RP)412 4170 y Fu(5)486 4206 y Fp(on)m(to)g(some)h (realization)g(of)f Fm(Q)p Fi(P)1683 4170 y Fu(5)1723 4206 y Fp(.)g(Both)h(steps)h(are)e(\(in)h(general\))f(not)h(natural)166 4327 y(for)25 b(di\013eren)m(tial)h(op)s(erators)f(in)g(the)h(sense)h (that,)e(in)g(order)h(to)f(p)s(erform)g(them,)h(one)f(needs)166 4447 y(some)33 b(additional)g(information.)166 4673 y(T)-8 b(o)42 b(o)m(v)m(ercome)h(these)g(di\016culties,)h(w)m(e)e(apply)g(the) g(metho)s(ds)h(dev)m(elop)s(ed)g(in)f(the)g(\014rst)166 4793 y(part)32 b(of)h(this)g(article)g([19])f(\(whic)m(h)i(w)m(e)g (will)f(call)g(P)m(art)g(I\).)166 5019 y(The)i(plan)e(of)g(the)h (article)h(is)f(the)g(follo)m(wing.)g(In)g(Section)g(2)f(w)m(e)i (describ)s(e)g(Dirac's)f(con-)166 5139 y(struction,)48 b(in)g(Section)g(3)f(w)m(e)h(discuss)h(the)f(dimensional)h(reduction)f (of)f Fm(\034)11 b Fp(\()p Fi(R)3172 5103 y Fu(6)3211 5139 y Fp(\))48 b(and)166 5259 y Fm(\034)219 5223 y Fk(\003)259 5259 y Fp(\()p Fi(R)369 5223 y Fu(6)408 5259 y Fp(\),)42 b(Sections)g(4)f(and)g(5)g(are)h(dev)m(oted)h(to)d(the)i(reduction)g (of)f(the)h(6-dimensional)166 5380 y(Maxw)m(ell)c(equations,)g(and)f (\014nally)-8 b(,)37 b(in)g(Section)g(6,)f(w)m(e)i(giv)m(e)f(a)f (complete)i(set)f(of)f(split-)1769 5712 y(2)p eop %%Page: 3 3 3 2 bop 166 83 a Fp(ting)33 b(relations)g(for)f(the)h(Dirac's)g (construction.)166 637 y Fq(2)112 b(General)39 b(Description)e(of)h (the)f(Dirac's)h(Construction)166 1000 y Fp(W)-8 b(e)24 b(start)f(with)h(the)g(conformal)g(compacti\014cation)p 2087 922 105 4 v 24 w Fm(M)38 b Fp(:=)28 b(\()p Fm(S)2454 964 y Fu(1)2497 1000 y Fl(\002)s Fm(S)2643 964 y Fu(3)2683 1000 y Fp(\))p Fm(=)p Fi(Z)2836 1015 y Fu(2)2899 1000 y Fp(of)23 b(Mink)m(o)m(wski)166 1120 y(space)52 b(\(for)f(details)g (see)i([21]\).)d(It)h(is)h(based)g(on)f(the)g(Dirac's)g(construction)i ([4])d(in)166 1241 y(whic)m(h)33 b(the)e(p)s(oin)m(ts)h(of)p 1011 1163 V 30 w Fm(M)42 b Fp(are)31 b(iden)m(ti\014ed)i(with)e(the)h (isotropic)f(straigh)m(t)h(lines)g(in)f(the)g(6-)166 1361 y(dimensional)23 b(real)f(space)h Fi(R)1193 1325 y Fu(6)1254 1361 y Fp(with)f(a)g(diagonal)f(metric)i(tensor)f(\()p Fm(\021)2576 1376 y Fh(mn)2686 1361 y Fp(\))27 b(=)h(diag)q(\()p Fl(\000)p Fp(1)p Fm(;)17 b Fp(1)p Fm(;)g Fp(1)p Fm(;)g Fp(1)p Fm(;)g Fp(1)p Fm(;)g Fl(\000)p Fp(1\).)166 1482 y(Let)35 b Fm(Q)420 1445 y Fu(6)494 1482 y Fp(b)s(e)g(the)g(ligh)m(t)g (cone)g(in)g Fi(R)1438 1445 y Fu(6)1512 1482 y Fp(and)f Fi(R)h Fp(b)s(e)g(the)g(m)m(ultiplicativ)m(e)i(group)d(of)g(nonzero)166 1602 y(real)e(n)m(um)m(b)s(ers)h(acting)e(on)h Fi(R)1245 1566 y Fu(6)1316 1602 y Fp(\(here)g(and)f(elsewhere,)j Fi(R)2284 1566 y Fu(6)2355 1602 y Fp(and)e Fm(Q)2621 1566 y Fu(6)2692 1602 y Fp(stand)g(for)e Fi(R)3176 1566 y Fu(6)3236 1602 y Fl(n)19 b(f)p Fp(0)p Fl(g)166 1722 y Fp(and)33 b Fm(Q)433 1686 y Fu(6)495 1722 y Fl(n)22 b(f)p Fp(0)p Fl(g)p Fp(,)32 b(resp)s(ectiv)m(ely\).)j(Therefore)p 1813 1644 V 34 w Fm(M)43 b Fp(is)33 b(the)g(pro)5 b(jected)34 b(ligh)m(t)f(cone:)p 1375 1909 V 1375 1988 a Fm(M)1507 1960 y Fl(\030)1508 1992 y Fp(=)1612 1988 y Fm(Q)p Fi(P)1750 1946 y Fu(5)1817 1988 y Fp(:=)28 b Fm(Q)2025 1946 y Fu(6)2065 1988 y Fm(=)p Fi(R)k Fm(:)166 2253 y Fp(Let)40 b(\()p Fm(\030)434 2217 y Fh(m)500 2253 y Fp(\))g(:=)g(\()p Fm(\030)807 2217 y Fh(\026)853 2253 y Fm(;)17 b(\030)945 2217 y Fu(5)983 2253 y Fm(;)g(\030)1075 2217 y Fu(6)1114 2253 y Fp(\))40 b(\()p Fm(m;)17 b(n;)g(:)g(:)g(:)39 b Fp(=)h(0)p Fm(;)17 b Fp(1)p Fm(;)g Fp(2)p Fm(;)g Fp(3)p Fm(;)g Fp(5)p Fm(;)g Fp(6;)37 b Fm(\026;)17 b(\027)q(;)g(:)g(:)g(:)38 b Fp(=)i(0)p Fm(;)17 b Fp(1)p Fm(;)g Fp(2)p Fm(;)g Fp(3\))38 b(b)s(e)i(the)166 2373 y(canonical)31 b(co)s(ordinates)g(in)f Fi(R)1291 2337 y Fu(6)1331 2373 y Fp(.)g(The)i(isomorphism)f Fm(Q)p Fi(P)2292 2337 y Fu(5)2360 2345 y Fl(\030)2361 2377 y Fp(=)2465 2373 y(\()p Fm(S)2569 2337 y Fu(1)2626 2373 y Fl(\002)18 b Fm(S)2787 2337 y Fu(3)2826 2373 y Fp(\))p Fm(=)p Fi(Z)2979 2388 y Fu(2)3049 2373 y Fp(is)31 b(eviden)m(t)166 2493 y(from)281 2759 y Fm(S)347 2717 y Fu(1)409 2759 y Fl(\002)22 b Fm(S)574 2717 y Fu(3)641 2759 y Fp(=)28 b Fl(f)16 b Fm(\030)32 b Fl(2)c Fi(R)1052 2717 y Fu(6)1136 2759 y Fl(j)44 b Fp(\()p Fm(\030)1294 2717 y Fu(0)1333 2759 y Fp(\))1371 2717 y Fu(2)1432 2759 y Fp(+)22 b(\()p Fm(\030)1616 2717 y Fu(6)1655 2759 y Fp(\))1693 2717 y Fu(2)1760 2759 y Fp(=)27 b(\()p Fm(\030)1949 2717 y Fu(1)1988 2759 y Fp(\))2026 2717 y Fu(2)2088 2759 y Fp(+)22 b(\()p Fm(\030)2272 2717 y Fu(2)2311 2759 y Fp(\))2349 2717 y Fu(2)2410 2759 y Fp(+)g(\()p Fm(\030)2594 2717 y Fu(3)2633 2759 y Fp(\))2671 2717 y Fu(2)2732 2759 y Fp(+)g(\()p Fm(\030)2916 2717 y Fu(5)2955 2759 y Fp(\))2993 2717 y Fu(2)3060 2759 y Fp(=)28 b(1)17 b Fl(g)32 b Fm(:)166 3146 y Fp(It)i(is)h(con)m(v)m(enien)m(t)i(to)d(in)m(tro)s(duce)h(in)f Fi(R)1600 3110 y Fu(6)1674 3146 y Fp(co)s(ordinates)h(adapted)f(to)g (the)h(ligh)m(t)f(cone)h Fm(Q)3387 3110 y Fu(6)3427 3146 y Fp(.)166 3266 y(F)-8 b(ollo)m(wing)33 b([17,10,7,13],)e(w)m(e)j(set) 849 3571 y Fm(x)904 3535 y Fh(\026)951 3571 y Fp(:=)1209 3521 y Fm(\030)1257 3480 y Fh(\026)p 1109 3547 294 4 v 1109 3635 a Fm(\030)1157 3594 y Fu(5)1218 3635 y Fp(+)22 b Fm(\030)1364 3594 y Fu(6)1805 3571 y Fm(\030)1853 3535 y Fh(\026)1927 3571 y Fp(=)27 b Fm(k)20 b(x)2156 3535 y Fh(\026)849 3822 y Fm(k)51 b Fp(:=)45 b Fm(\030)1147 3786 y Fu(5)1208 3822 y Fp(+)22 b Fm(\030)1354 3786 y Fu(6)1805 3822 y Fm(\030)1853 3786 y Fu(5)1919 3822 y Fp(=)28 b Fm(k)2110 3701 y Fg(\022)2181 3783 y Fp(1)22 b Fl(\000)h Fm(x)2407 3742 y Fu(2)p 2181 3799 266 4 v 2290 3866 a Fp(2)2479 3822 y(+)f Fm(\036)2635 3701 y Fg(\023)849 4050 y Fm(\036)44 b Fp(:=)1226 4001 y Fm(\030)1274 3960 y Fh(m)1340 4001 y Fm(\030)1383 4016 y Fh(m)p 1109 4027 458 4 v 1109 4114 a Fp(2\()p Fm(\030)1244 4073 y Fu(5)1304 4114 y Fp(+)22 b Fm(\030)1450 4073 y Fu(6)1489 4114 y Fp(\))1527 4073 y Fu(2)1805 4050 y Fm(\030)1853 4014 y Fu(6)1919 4050 y Fp(=)28 b Fm(k)2110 3929 y Fg(\022)2181 4011 y Fp(1)22 b(+)g Fm(x)2405 3970 y Fu(2)p 2181 4027 264 4 v 2289 4094 a Fp(2)2477 4050 y(+)g Fm(\036)2633 3929 y Fg(\023)2743 4050 y Fm(;)166 4371 y Fp(where)34 b Fm(x)503 4386 y Fh(\026)578 4371 y Fp(:=)27 b Fm(\021)756 4386 y Fh(\026\027)842 4371 y Fm(x)897 4335 y Fh(\027)941 4371 y Fp(,)32 b Fm(\021)1048 4386 y Fh(\026\027)1162 4371 y Fp(:=)27 b(diag)q(\()p Fl(\000)p Fp(1)p Fm(;)17 b Fp(1)p Fm(;)g Fp(1)p Fm(;)g Fp(1\),)32 b Fm(x)2067 4335 y Fu(2)2134 4371 y Fp(:=)c Fm(x)2320 4335 y Fh(\026)2367 4371 y Fm(x)2422 4386 y Fh(\026)2469 4371 y Fp(.)33 b(The)g(Jacobian)g (is)1385 4636 y Fm(@)5 b Fp(\()p Fm(\030)1527 4595 y Fh(\026)1574 4636 y Fm(;)17 b(\030)1666 4595 y Fu(5)1704 4636 y Fm(;)g(\030)1796 4595 y Fu(6)1835 4636 y Fp(\))p 1385 4680 488 4 v 1412 4772 a Fm(@)5 b Fp(\()p Fm(x)1561 4730 y Fh(\026)1609 4772 y Fm(;)17 b(k)s(;)g(\036)p Fp(\))1911 4703 y(=)27 b Fl(\000)p Fm(k)2145 4662 y Fu(2)2218 4703 y Fm(;)166 5019 y Fp(hence)32 b(this)f(c)m(hange)g(of)f(v)-5 b(ariables)31 b(is)g(nondegenerate)h(outside)g(the)e(h)m(yp)s(erplane)j Fm(k)d Fp(=)e(0.)166 5139 y(The)38 b(co)s(ordinates)f(\()p Fm(x)986 5103 y Fh(\026)1033 5139 y Fm(;)17 b(k)s(;)g(\036)p Fp(\))36 b(will)i(b)s(e)f(referred)h(to)e(as)h Fm(Q)p Fp(-co)s(ordinates)h(\(the)f(equation)166 5259 y(of)28 b(the)g(ligh)m(t)h(cone)g Fm(Q)952 5223 y Fu(6)1019 5259 y Fp(in)g Fm(Q)p Fp(-co)s(ordinates)f(is)h(2)p Fm(k)1949 5223 y Fu(2)1988 5259 y Fm(\036)f Fp(=)f(0\),)h(whereas)i(\()p Fm(\030)2770 5223 y Fh(m)2836 5259 y Fp(\))e(will)g(b)s(e)h(called)166 5380 y Fm(\030)5 b Fp(-co)s(ordinates.)45 b(W)-8 b(e)47 b(use)f(the)h(indices)g(\\)p Fl(\000)p Fp(")f(for)f Fm(k)s Fp(,)h(\\+")f(for)h Fm(\036)p Fp(,)f(and)h Fm(M)5 b(;)17 b(N)5 b(;)17 b(:)g(:)g(:)45 b Fp(for)1769 5712 y(3)p eop %%Page: 4 4 4 3 bop 166 83 a Fp(\()p Fm(\026;)17 b Fl(\000)p Fm(;)g Fp(+\),)32 b(etc.)h(In)g Fm(Q)p Fp(-co)s(ordinates)g(the)g(comp)s(onen) m(ts)h(of)e(the)h(metric)h(tensor)f(are)861 520 y(\()p Fm(g)946 535 y Fh(M)7 b(N)1088 520 y Fp(\))27 b(:=)1284 225 y Fg(0)1284 371 y(B)1284 421 y(B)1284 470 y(B)1284 520 y(B)1284 570 y(B)1284 623 y(@)1379 351 y Fm(g)1426 366 y Fh(\026\027)1557 351 y Fm(g)1604 366 y Fh(\026)p Fk(\000)1751 351 y Fm(g)1798 366 y Fh(\026)p Fu(+)1373 531 y Fm(g)1420 546 y Fk(\000)p Fh(\027)1551 531 y Fm(g)1598 546 y Fk(\000\000)1745 531 y Fm(g)1792 546 y Fk(\000)p Fu(+)1373 712 y Fm(g)1420 727 y Fu(+)p Fh(\027)1551 712 y Fm(g)1598 727 y Fu(+)p Fk(\000)1745 712 y Fm(g)1792 727 y Fu(++)1922 225 y Fg(1)1922 371 y(C)1922 421 y(C)1922 470 y(C)1922 520 y(C)1922 570 y(C)1922 623 y(A)2023 520 y Fp(=)2126 225 y Fg(0)2126 371 y(B)2126 421 y(B)2126 470 y(B)2126 520 y(B)2126 570 y(B)2126 623 y(@)2215 351 y Fm(k)2269 315 y Fu(2)2309 351 y Fm(\021)2357 366 y Fh(\026\027)2505 351 y Fp(0)64 b(0)2305 531 y(0)122 b(2)p Fm(\036)33 b(k)2305 712 y Fp(0)148 b Fm(k)65 b Fp(0)2686 225 y Fg(1)2686 371 y(C)2686 421 y(C)2686 470 y(C)2686 520 y(C)2686 570 y(C)2686 623 y(A)166 965 y Fp(and)33 b(the)g(nonzero)g(Christo\013el)h(sym)m(b)s(ols)g(are)e(\000)1957 918 y Fh(\026)1957 984 y(\027)t Fk(\000)2083 965 y Fp(=)2198 926 y Fu(1)p 2197 942 39 4 v 2197 999 a Fh(k)2262 965 y Fm(\021)2314 929 y Fh(\026)2310 989 y(\027)2360 965 y Fp(,)h(\000)2481 929 y Fu(+)2481 989 y Fh(\026\027)2594 965 y Fp(=)27 b Fl(\000)p Fm(\021)2822 980 y Fh(\026\027)2909 965 y Fp(,)32 b(\000)3029 929 y Fu(+)3029 989 y Fk(\000)p Fu(+)3171 965 y Fp(=)3286 926 y Fu(1)p 3284 942 V 3284 999 a Fh(k)3333 965 y Fp(.)166 1188 y(The)i(action)e(of)g(the)h(group)g Fi(R)f Fp(on)h Fi(R)1527 1151 y Fu(6)1599 1188 y Fp(in)f Fm(Q)p Fp(-co)s(ordinates)h(is)1287 1413 y Fm(\032)p Fp(\()p Fm(x)1430 1372 y Fh(\026)1477 1413 y Fm(;)17 b(k)s(;)g(\036)p Fp(\))27 b(=)h(\()p Fm(x)1939 1372 y Fh(\026)1986 1413 y Fm(;)17 b(\032k)s(;)g(\036)p Fp(\))32 b Fm(;)166 1639 y Fp(so)h(as)g(a)f(c)m(hart)h(of)f Fi(RP)983 1602 y Fu(5)1044 1639 y Fl(n)22 b(f)p Fm(k)31 b Fp(=)c(0)p Fl(g)33 b Fp(w)m(e)g(use)h(the)f(h)m(yp)s(erplane)983 1864 y Fm(U)38 b Fp(:=)28 b Fl(f)p Fm(\030)k Fl(2)c Fi(R)1509 1823 y Fu(6)1576 1864 y Fl(j)f Fm(k)k Fp(=)c(1)p Fl(g)1943 1836 y(\030)1943 1868 y Fp(=)2048 1864 y Fi(RP)2181 1823 y Fu(5)2242 1864 y Fl(n)22 b(f)p Fm(k)31 b Fp(=)c(0)p Fl(g)32 b Fm(;)622 b Fp(\(2\))166 2192 y(transv)m(ersal)30 b(to)e(the)h Fi(R)p Fp(-orbits)f(of)g(the)h(p)s(oin)m(ts)g(of)f Fi(R)2051 2156 y Fu(6)2104 2192 y Fl(n)14 b(f)p Fm(k)30 b Fp(=)e(0)p Fl(g)p Fp(.)g(The)h(image)g(of)f(the)h(line)166 2312 y([\()p Fm(x)286 2276 y Fh(\026)333 2312 y Fm(;)17 b(k)s(;)g(\036)p Fp(\)])32 b(through)h(the)h(p)s(oin)m(t)f(\()p Fm(x)1516 2276 y Fh(\026)1563 2312 y Fm(;)17 b(k)s(;)g(\036)p Fp(\),)32 b Fm(k)f Fl(6)p Fp(=)e(0)j(\(i.e.,)i(of)e(the)i(equiv)-5 b(alence)35 b(class)f(of)166 2433 y(this)f(p)s(oin)m(t)g(with)g(resp)s (ect)h(to)e(the)h(action)g(of)f Fi(R)p Fp(\))g(is)h(the)g(p)s(oin)m(t)g (\()p Fm(x)2614 2397 y Fh(\026)2661 2433 y Fm(;)17 b Fp(1)p Fm(;)g(\036)p Fp(\))27 b Fl(2)h Fm(U)10 b Fp(.)166 2655 y(The)39 b(p)s(oin)m(ts)g(of)e(the)i(conformal)f (compacti\014cation)p 2156 2577 105 4 v 39 w Fm(M)48 b Fp(of)38 b(Mink)m(o)m(wski)j(space)e(are)f(re-)166 2776 y(alized)h(as)g(elemen)m(ts)i(of)c(the)i(pro)5 b(jected)40 b(ligh)m(t)f(cone)g Fm(Q)p Fi(P)2303 2740 y Fu(5)2342 2776 y Fp(,)g(i.e.,)g(as)f(isotropic)h(straigh)m(t)166 2896 y(lines)33 b(in)f Fi(R)573 2860 y Fu(6)612 2896 y Fp(,)g(and)f(the)h(p)s(oin)m(ts)h(of)e(Mink)m(o)m(wski)j(space)f Fm(M)42 b Fp(corresp)s(ond)33 b(to)e(the)h(isotropic)166 3017 y(straigh)m(t)h(lines)h(in)f Fi(R)936 2980 y Fu(6)1007 3017 y Fp(lying)h(outside)f(the)g(h)m(yp)s(erplane)h Fm(k)d Fp(=)d(0.)k(W)-8 b(e)33 b(use)h(the)f(manifold)806 3242 y Fm(M)38 b Fp(:=)28 b Fl(f)16 b Fm(\030)32 b Fl(2)c Fi(R)1376 3201 y Fu(6)1460 3242 y Fl(j)44 b Fm(k)31 b Fp(=)c(1)p Fm(;)49 b(\036)28 b Fp(=)f(0)17 b Fl(g)2174 3214 y(\030)2175 3246 y Fp(=)2279 3242 y Fm(Q)p Fi(P)2417 3201 y Fu(5)2479 3242 y Fl(n)22 b(f)p Fm(k)30 b Fp(=)e(0)p Fl(g)444 b Fp(\(3\))166 3570 y(and)33 b(the)g(mapping)561 3796 y Fm(\037)28 b Fp(:)f Fm(Q)p Fi(P)842 3754 y Fu(5)904 3796 y Fl(n)22 b(f)p Fm(k)31 b Fp(=)c(0)p Fl(g)g(\000)-16 b(!)28 b Fm(M)38 b Fp(:)28 b([\()p Fm(x)1833 3754 y Fh(\026)1880 3796 y Fm(;)17 b(k)s(;)g(\036)p Fp(\)])27 b Fl(7\000)-16 b(!)27 b Fp(\()p Fm(x)2453 3754 y Fh(\026)2500 3796 y Fm(;)17 b Fp(1)p Fm(;)g Fp(0\))26 b(=:)i(\()p Fm(x)2974 3754 y Fh(\026)3021 3796 y Fp(\))166 4021 y(as)33 b(a)f(c)m(hart)h(of)f Fm(Q)p Fi(P)868 3985 y Fu(5)930 4021 y Fl(n)22 b(f)p Fm(k)30 b Fp(=)e(0)p Fl(g)p Fp(.)166 4244 y(In)k(the)g(Dirac's)g (construction)h(the)f(isomorphism)h Fm(C)2147 4259 y Fu(0)2186 4244 y Fp(\(1)p Fm(;)17 b Fp(3\))2431 4216 y Fl(\030)2432 4248 y Fp(=)2536 4244 y Fm(O)2611 4259 y Fu(0)2650 4244 y Fp(\(2)p Fm(;)g 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Fp(and)h(the)g(corresp)s(onding)g(fundamen)m(tal)h(v)m(ector)f (\014elds)h(are)609 5325 y Fm(X)690 5340 y Fh(mn)827 5325 y Fp(=)28 b(\()p Fm(X)1050 5340 y Fh(mn)1159 5325 y Fp(\))1197 5284 y Fh(p)1237 5340 y(q)1292 5325 y Fm(\030)1340 5284 y Fh(q)1447 5258 y Fm(@)p 1404 5302 145 4 v 1404 5394 a(@)5 b(\030)1508 5353 y Fh(p)1585 5325 y Fp(=)28 b Fm(\030)1732 5340 y Fh(m)1872 5258 y Fm(@)p 1825 5302 152 4 v 1825 5394 a(@)5 b(\030)1929 5353 y Fh(n)2008 5325 y Fl(\000)23 b Fm(\030)2151 5340 y Fh(n)2281 5258 y Fm(@)p 2224 5302 171 4 v 2224 5394 a(@)5 b(\030)2328 5353 y Fh(m)2437 5325 y Fm(;)212 b(m)28 b(<)f(n)33 b(:)1769 5712 y Fp(4)p eop %%Page: 5 5 5 4 bop 166 83 a Fp(The)34 b(\\ph)m(ysical)g(fundamen)m(tal)f(v)m (ector)h(\014elds")g([20])e(are)388 484 y Fm(M)482 499 y Fh(\026\027)568 484 y Fp(:=)c Fm(X)780 499 y Fh(\026\027)1224 484 y Fp(=)f Fm(x)1382 499 y Fh(\026)1429 484 y Fm(@)1480 499 y Fh(\027)1546 484 y Fl(\000)c Fm(x)1701 499 y Fh(\027)1744 484 y Fm(@)1795 499 y Fh(\026)1875 484 y Fm(;)407 b Fp(0)27 b Fl(\024)h Fm(\026)g(<)f(\027)34 b Fl(\024)29 b Fp(3)388 700 y Fm(P)451 715 y Fh(\026)568 700 y Fp(:=)f Fm(X)780 715 y Fh(\026)p Fu(6)884 700 y Fl(\000)22 b Fm(X)1064 715 y Fh(\026)p Fu(5)1224 700 y Fp(=)27 b Fm(@)1378 715 y Fh(\026)388 916 y Fm(K)471 931 y Fh(\026)568 916 y Fp(:=)h Fl(\000)p Fm(X)857 931 y Fh(\026)p Fu(5)961 916 y Fl(\000)23 b Fm(X)1142 931 y Fh(\026)p Fu(6)1224 916 y Fp(=)k(2)p Fm(x)1431 931 y Fh(\026)1478 916 y Fp(\()p Fm(x)22 b Fl(\001)g Fm(@)5 b Fp(\))23 b(+)f(\(2)p Fm(\036)g Fl(\000)g Fm(x)2179 880 y Fu(2)2219 916 y Fp(\))p Fm(@)2308 931 y Fh(\026)2378 916 y Fl(\000)g Fp(2)p Fm(k)s(x)2635 931 y Fh(\026)2682 916 y Fm(@)2733 931 y Fk(\000)2815 916 y Fp(+)g(4)p Fm(\036x)3075 931 y Fh(\026)3121 916 y Fm(@)3172 931 y Fu(+)388 1132 y Fm(D)99 b Fp(:=)28 b Fl(\000)p Fm(X)857 1147 y Fu(56)1224 1132 y Fp(=)f Fm(x)c Fl(\001)f Fm(@)28 b Fl(\000)22 b Fm(k)s(@)1738 1147 y Fk(\000)1820 1132 y Fl(\000)h Fp(2)p Fm(\036@)2078 1147 y Fu(+)2169 1132 y Fm(;)166 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2624 y Fp(\()p Fm(\030)5 b(;)17 b(w)s Fp(\))26 b(:=)i(\()p Fm(\032\030)5 b(;)17 b(\032)2110 2583 y Fk(\000)p Fh(\025)2209 2624 y Fm(w)s Fp(\))32 b Fm(;)950 b Fp(\(8\))166 2946 y(where)36 b(\()p Fm(\030)5 b(;)17 b(w)s Fp(\))31 b Fl(2)h Fm(\034)873 2910 y Fk(\003)913 2946 y Fp(\()p Fi(R)1023 2910 y Fu(6)1063 2946 y Fp(\))1101 2975 y Fh(\030)1138 2946 y Fp(.)k(An)f Fi(R)p Fp(-in)m(v)-5 b(arian)m(t)35 b(di\013eren)m(tial)i(1-form)d Fl(A)d(2)i Fm(C)2971 2910 y Fk(1)3046 2946 y Fp(\()p Fm(\034)3137 2910 y Fk(\003)3176 2946 y Fp(\()p Fi(R)3286 2910 y Fu(6)3326 2946 y Fp(\)\))3402 2961 y Ff(R)166 3079 y Fp(satis\014es)h Fl(A)605 3094 y Fh(m)671 3079 y Fp(\()p Fm(\032\030)5 b Fp(\))28 b(=)f Fm(\032)1026 3043 y Fk(\000)p Fh(\025)1127 3079 y Fl(A)1207 3094 y Fh(m)1273 3079 y Fp(\()p Fm(\030)5 b Fp(\),)32 b(whic)m(h)i(in)f Fm(Q)p Fp(-co)s(ordinates)g(reads)361 3410 y Fm(A)434 3425 y Fh(\026;)12 b Fu(+)567 3410 y Fp(\()p Fm(x;)17 b(\032k)s(;)g(\036)p Fp(\))g(=)f Fm(\032)1107 3369 y Fu(1)p Fk(\000)p Fh(\025)1243 3410 y Fm(A)1316 3425 y Fh(\026;)c Fu(+)1449 3410 y Fp(\()p Fm(x;)17 b(k)s(;)g(\036)p Fp(\))435 3558 y Fm(A)508 3573 y Fk(\000)567 3558 y Fp(\()p Fm(x;)g(\032k)s(;)g(\036)p Fp(\))g(=)f Fm(\032)1107 3517 y Fk(\000)p Fh(\025)1208 3558 y Fm(A)1281 3573 y Fk(\000)1340 3558 y Fp(\()p Fm(x;)h(k)s(;)g(\036)p Fp(\))32 b Fm(:)166 3881 y Fp(In)f Fm(Q)p Fp(-co)s(ordinates,)g(the)f(Loren)m(tz)i (transformations)e(do)h(not)f(c)m(hange)h Fm(A)2857 3896 y Fk(\000)2947 3881 y Fp(and)f Fm(A)3207 3896 y Fu(+)3297 3881 y Fp(and)166 4001 y(act)h(on)f Fm(A)533 4016 y Fh(\026)610 4001 y Fp(in)h(the)g(usual)g(w)m(a)m(y;)h(the)f(translations)g(do)f (not)h(c)m(hange)g(an)m(y)g(of)f(the)h(comp)s(o-)166 4122 y(nen)m(ts)24 b(of)e Fl(A)p Fp(;)g(the)h(actions)g(of)f(the)g(sp)s (ecial)i(conformal)e(transformations)h(and)f(dilatations)166 4242 y(are)33 b(giv)m(en)g(in)g(App)s(endix)h(B.)166 4463 y(In)i(the)h(case)g(of)e Fm(\034)11 b Fp(\()p Fm(Q)955 4427 y Fu(6)996 4463 y Fp(\),)36 b(w)m(e)h(\014rst)f(p)s(erform)g Fi(R)p Fp(-dimensional)h(reduction,)g(whic)m(h)h(yields)166 4583 y(the)33 b Fm(O)409 4598 y Fu(0)448 4583 y Fp(\(2)p Fm(;)17 b Fp(4\))665 4612 y Ff(R)717 4583 y Fp(-in)m(tert)m(wining)34 b(SES)640 4836 y(0)48 b Fl( )-30 b(\000)-22 b(\000)-30 b(\000)49 b Fm(\034)1088 4800 y Fu(v)1131 4836 y Fp(\()p Fi(R)1241 4800 y Fu(6)1280 4836 y Fp(\))1318 4800 y Fk(\003)1318 4861 y Fh(U)1521 4783 y(i)1545 4759 y Fd(\003)1426 4836 y Fl( )-30 b(\000)-22 b(\000)-30 b(\000)49 b Fm(\034)1777 4800 y Fk(\003)1817 4836 y Fp(\()p Fi(R)1927 4800 y Fu(6)1966 4836 y Fp(\))2004 4851 y Fh(U)2202 4777 y(j)2235 4754 y Fd(\003)2112 4836 y Fl( )-30 b(\000)-23 b(\000)-30 b(\000)50 b Fm(\034)2463 4800 y Fk(\003)2503 4836 y Fp(\()p Fm(U)10 b Fp(\))49 b Fl( )-30 b(\000)-23 b(\000)-30 b(\000)50 b Fp(0)3329 4816 y(\(9\))166 5139 y(of)e(v)m(ector)i(bundles)g(o)m(v)m (er)g Fm(U)10 b Fp(.)49 b(In)g(the)h(pro)s(cess)f(of)g(reduction,)g(w)m (e)h(ha)m(v)m(e)g(used)g(that)166 5259 y(the)d(action)f(of)g Fi(R)g Fp(on)g Fi(R)1118 5223 y Fu(6)1204 5259 y Fp(is)g(free,)h(hence) h(\()p Fm(\034)1919 5223 y Fu(v)1962 5259 y Fp(\()p Fi(R)2072 5223 y Fu(6)2111 5259 y Fp(\))2149 5223 y Fk(\003)2188 5259 y Fp(\))2226 5274 y Ff(R)2329 5259 y Fp(=)j(\()p Fm(\034)2547 5223 y Fu(v)2590 5259 y Fp(\()p Fi(R)2700 5223 y Fu(6)2739 5259 y Fp(\))2777 5274 y Ff(R)2829 5259 y Fp(\))2867 5223 y Fk(\003)2958 5259 y Fp(=)g Fm(\034)3138 5223 y Fu(v)3180 5259 y Fp(\()p Fi(R)3290 5223 y Fu(6)3330 5259 y Fp(\))3368 5223 y Fk(\003)3368 5284 y Fh(U)3427 5259 y Fp(,)166 5380 y Fm(\034)219 5344 y Fk(\003)259 5380 y Fp(\()p Fi(R)369 5344 y Fu(6)408 5380 y Fp(\))446 5395 y Ff(R)526 5380 y Fp(=)28 b Fm(\034)683 5344 y Fk(\003)722 5380 y Fp(\()p Fi(R)832 5344 y Fu(6)872 5380 y Fp(\))910 5395 y Fh(U)969 5380 y Fp(,)k(and)h(\()p Fm(p)1305 5344 y Fk(\003)1344 5380 y Fm(\034)1397 5344 y Fk(\003)1437 5380 y Fp(\()p Fi(RP)1608 5344 y Fu(5)1647 5380 y Fp(\)\))1723 5395 y Ff(R)1803 5380 y Fp(=)28 b Fm(\034)1960 5344 y Fk(\003)2000 5380 y Fp(\()p Fm(U)10 b Fp(\).)1769 5712 y(7)p eop %%Page: 8 8 8 7 bop 166 83 a Fp(Restricting)34 b(\(9\))f(to)h(the)f Fm(O)1189 98 y Fu(0)1228 83 y Fp(\(2)p Fm(;)17 b Fp(4\))1445 112 y Ff(R)1498 83 y Fp(-in)m(v)-5 b(arian)m(t)33 b(submanifold)i Fm(M)40 b Fl(\032)29 b Fm(U)10 b Fp(,)34 b(w)m(e)h(obtain)e(the)166 203 y(follo)m(wing)g Fm(C)648 218 y Fu(0)687 203 y Fp(\(1)p Fm(;)17 b Fp(3\))o(-in)m(tert)m(wining)34 b(SES)f(of)g(v)m(ector)g (bundles)h(o)m(v)m(er)g Fm(M)526 457 y Fp(0)49 b Fl( )-30 b(\000)-23 b(\000)-30 b(\000)49 b Fm(\034)974 421 y Fu(v)1017 457 y Fp(\()p Fi(R)1127 421 y Fu(6)1167 457 y Fp(\))1205 421 y Fk(\003)1205 482 y Fh(M)1427 404 y(i)1451 381 y Fd(\003)1332 457 y Fl( )-30 b(\000)-22 b(\000)-30 b(\000)49 b Fm(\034)1683 421 y Fk(\003)1723 457 y Fp(\()p Fi(R)1833 421 y Fu(6)1872 457 y Fp(\))1910 472 y Fh(M)2128 399 y(j)2161 375 y Fd(\003)2038 457 y Fl( )-30 b(\000)-22 b(\000)-30 b(\000)49 b Fm(\034)2389 421 y Fk(\003)2429 457 y Fp(\()p Fm(U)10 b Fp(\))2581 472 y Fh(M)2709 457 y Fl( )-30 b(\000)-22 b(\000)-30 b(\000)49 b Fp(0)32 b Fm(:)3280 438 y Fp(\(10\))166 780 y(On)e(the)g(other)f(hand,)h(the)g Fm(O)1250 795 y Fu(0)1289 780 y Fp(\(2)p Fm(;)17 b Fp(4\))1506 809 y Ff(R)1558 780 y Fp(-in)m(v)-5 b(ariance)30 b(of)f Fm(M)41 b Fp(as)29 b(a)h(submanifold)g(of)f Fm(U)40 b Fp(giv)m(es)166 900 y(us)33 b(the)g Fm(C)529 915 y Fu(0)568 900 y Fp(\(1)p Fm(;)17 b Fp(3\)-in)m(tert)m(wining)33 b(SES)613 1151 y(0)48 b Fl( )-30 b(\000)-22 b(\000)-30 b(\000)49 b Fm(\034)1061 1115 y Fk(\003)1101 1151 y Fp(\()p Fm(M)10 b Fp(\))1418 1098 y Fh(k)1457 1075 y Fd(\003)1330 1151 y Fl( )-30 b(\000)-22 b(\000)-30 b(\000)49 b Fm(\034)1681 1115 y Fk(\003)1721 1151 y Fp(\()p Fm(U)10 b Fp(\))1873 1166 y Fh(M)2097 1098 y(l)2119 1075 y Fd(\003)2001 1151 y Fl( )-30 b(\000)-22 b(\000)-30 b(\000)49 b Fm(\027)2353 1115 y Fk(\003)2393 1151 y Fp(\()p Fm(M)10 b Fp(\))50 b Fl( )-30 b(\000)-23 b(\000)-30 b(\000)49 b Fp(0)33 b Fm(;)3280 1135 y Fp(\(11\))166 1476 y(where)23 b Fm(k)31 b Fp(:)d Fm(\034)11 b Fp(\()p Fm(M)f Fp(\))28 b Fl(!)g Fm(\034)11 b Fp(\()p Fm(U)f Fp(\))1168 1491 y Fh(M)1269 1476 y Fp(is)23 b(the)f(natural)f(em)m(b)s(edding)j(and)e Fm(\027)2560 1440 y Fk(\003)2599 1476 y Fp(\()p Fm(M)10 b Fp(\))29 b(:=)e(\()p Fm(\034)11 b Fp(\()p Fm(U)f Fp(\))3181 1491 y Fh(M)3261 1476 y Fm(=\034)h Fp(\()p Fm(M)f Fp(\)\))3581 1440 y Fk(\003)166 1596 y Fp(\(cf.)33 b(Eq.)g(\(7\))f(in)h(P)m(art)g (I\)\).)166 1824 y(F)-8 b(rom)44 b(\(10\))h(and)g(\(11\),)f(w)m(e)i (obtain)f(the)g(follo)m(wing)g Fm(C)2246 1839 y Fu(0)2285 1824 y Fp(\(1)p Fm(;)17 b Fp(3\))o(-in)m(v)-5 b(arian)m(t)45 b(\014ltration)g(of)166 1944 y(v)m(ector)34 b(bundles)g(o)m(v)m(er)g Fm(M)10 b Fp(:)878 2179 y Fm(j)924 2138 y Fk(\003)964 2179 y Fp(\()p Fm(l)1033 2138 y Fk(\003)1072 2179 y Fp(\()p Fm(\027)1164 2138 y Fk(\003)1204 2179 y Fp(\()p Fm(M)g Fp(\)\)\))56 b Fl(\032)g Fm(j)1695 2138 y Fk(\003)1734 2179 y Fp(\()p Fm(\034)1825 2138 y Fk(\003)1865 2179 y Fp(\()p Fm(U)10 b Fp(\))2017 2194 y Fh(M)2097 2179 y Fp(\))55 b Fl(\032)h Fm(\034)2376 2138 y Fk(\003)2416 2179 y Fp(\()p Fi(R)2526 2138 y Fu(6)2565 2179 y Fp(\))2603 2194 y Fh(M)2714 2179 y Fm(;)166 2413 y Fp(or,)32 b(in)h Fm(Q)p Fp(-co)s(ordinates,)732 2648 y Fl(f)p Fp(\(0)p Fm(;)17 b Fp(0)p Fm(;)g(A)1079 2663 y Fu(+)1137 2648 y Fp(\))p Fl(g)55 b(\032)h(f)p Fp(\()p Fm(A)1574 2663 y Fh(\026)1620 2648 y Fm(;)17 b Fp(0)p Fm(;)g(A)1830 2663 y Fu(+)1889 2648 y Fp(\))p Fl(g)55 b(\032)h(f)o Fp(\()p Fm(A)2325 2663 y Fh(\026)2372 2648 y Fm(;)17 b(A)2489 2663 y Fk(\000)2548 2648 y Fm(;)g(A)2665 2663 y Fu(+)2724 2648 y Fp(\))p Fl(g)49 b Fm(:)166 2882 y Fp(This)34 b(allo)m(ws)f(us)g(to)g(imp)s(ose)g(the)g(conformally)g(in)m (v)-5 b(arian)m(t)33 b(conditions)1145 3117 y Fm(A)1218 3132 y Fk(\000)1304 3117 y Fp(=)28 b(0)195 b(or)g Fm(A)2007 3132 y Fh(\026)2081 3117 y Fp(=)28 b Fm(A)2258 3132 y Fk(\000)2344 3117 y Fp(=)g(0)783 b(\(12\))166 3459 y(\(whic)m(h)34 b(can)f(b)s(e)g(noticed)g(from)f(the)h(transformation)g(la)m(ws)g(giv)m (en)h(in)f(App)s(endix)h(B\).)166 3909 y Fq(4)112 b(Dimensional)31 b(Reduction)e(of)g(the)g(6-Dimensional)i(Maxw)m(ell)f(Equations)166 4256 y Fp(Let)38 b Fl(A)e(2)h Fm(C)642 4220 y Fk(1)717 4256 y Fp(\()p Fm(\034)808 4220 y Fk(\003)848 4256 y Fp(\()p Fi(R)958 4220 y Fu(6)997 4256 y Fp(\)\))g(b)s(e)h(the)h (6-dimensional)f(electromagnetic)h(v)m(ector)g(p)s(oten)m(tial,)166 4377 y Fl(J)46 b(2)28 b Fm(C)451 4341 y Fk(1)525 4377 y Fp(\()p Fm(\034)616 4341 y Fk(\003)656 4377 y Fp(\()p Fi(R)766 4341 y Fu(6)806 4377 y Fp(\)\))k(b)s(e)h(the)g (electromagnetic)h(curren)m(t)g(densit)m(y)g(and)1155 4661 y Fl(F)1227 4676 y Fh(mn)1363 4661 y Fp(=)28 b(\(d)p Fl(A)p Fp(\))1677 4676 y Fh(mn)1814 4661 y Fp(=)1927 4593 y Fm(@)5 b Fl(A)2063 4608 y Fh(n)p 1927 4638 184 4 v 1933 4729 a Fm(@)g(\030)2037 4700 y Fh(m)2143 4661 y Fl(\000)2252 4593 y Fm(@)g Fl(A)2388 4608 y Fh(m)p 2252 4638 204 4 v 2278 4729 a Fm(@)g(\030)2382 4700 y Fh(n)166 4946 y Fp(b)s(e)22 b(the)g(\014eld)h(strength)f(tensor.)h (Then)f(the)h(6-dimensional)f(Maxw)m(ell)i(equations)f(\(6DME\))166 5066 y(are)1032 5258 y Fm(@)5 b Fl(F)1160 5273 y Fh(mn)p 1032 5302 238 4 v 1068 5394 a Fm(@)g(\030)1167 5409 y Fh(m)1308 5325 y Fp(=)27 b Fc(\003)1488 5340 y Fu(6)1528 5325 y Fl(A)1608 5340 y Fh(n)1677 5325 y Fl(\000)1834 5258 y Fm(@)p 1786 5302 152 4 v 1786 5394 a(@)5 b(\030)1890 5365 y Fh(n)1964 5325 y Fp(\()p Fl(r)2085 5340 y Fu(6)2147 5325 y Fl(\001)22 b(A)p Fp(\))27 b(=)g Fl(J)2513 5340 y Fh(n)2592 5325 y Fm(:)661 b Fp(\(13\))1769 5712 y(8)p eop %%Page: 9 9 9 8 bop 166 83 a Fp(Note)24 b(that)f(since)i(the)f(homogeneous)h(Maxw)m (ell)g(equations)g(d)p Fl(F)38 b Fp(=)27 b(0)d(are)f(automatically)166 203 y(conformally)38 b(in)m(v)-5 b(arian)m(t,)37 b(w)m(e)i(consider)f (only)g(the)f(inhomogeneous)i(ones)e(and)h(for)e(the)166 324 y(sak)m(e)e(of)e(brevit)m(y)i(w)m(e)g(omit)f(the)g(adjectiv)m(e)h (\\inhomogeneous".)166 558 y(In)f Fm(Q)p Fp(-co)s(ordinates,)g(\(13\))f (read)374 977 y Fl(r)457 936 y Fh(N)524 977 y Fm(F)587 992 y Fh(N)7 b(\026)713 977 y Fp(=)838 910 y(1)p 816 954 94 4 v 816 1045 a Fm(k)870 1017 y Fu(2)936 977 y Fm(@)992 936 y Fh(\027)1036 977 y Fm(F)1099 992 y Fh(\027)t(\026)1207 977 y Fl(\000)1319 910 y Fp(1)p 1316 954 55 4 v 1316 1045 a Fm(k)1397 977 y(@)1448 992 y Fu(+)1508 977 y Fm(F)1571 992 y Fh(\026)p Fk(\000)1694 977 y Fl(\000)1807 910 y Fp(1)p 1804 954 V 1804 1045 a Fm(k)1885 977 y(@)1936 992 y Fk(\000)1995 977 y Fm(F)2058 992 y Fh(\026)p Fu(+)2182 977 y Fp(+)2290 910 y(2)p Fm(\036)p 2290 954 107 4 v 2296 1045 a(k)2350 1017 y Fu(2)2423 977 y Fm(@)2474 992 y Fu(+)2534 977 y Fm(F)2597 992 y Fh(\026)p Fu(+)2726 977 y Fp(=)27 b Fm(J)2883 992 y Fh(\026)361 1215 y Fl(r)444 1174 y Fh(N)511 1215 y Fm(F)574 1230 y Fh(N)7 b Fk(\000)713 1215 y Fp(=)838 1148 y(1)p 816 1192 94 4 v 816 1284 a Fm(k)870 1255 y Fu(2)936 1215 y Fm(@)992 1174 y Fh(\027)1036 1215 y Fm(F)1099 1230 y Fh(\027)t Fk(\000)1219 1215 y Fl(\000)1331 1148 y Fp(1)p 1329 1192 55 4 v 1329 1284 a Fm(k)1409 1215 y(@)1460 1230 y Fk(\000)1520 1215 y Fm(F)1583 1230 y Fk(\000)p Fu(+)1719 1215 y Fp(+)1827 1148 y(2)p Fm(\036)p 1827 1192 107 4 v 1833 1284 a(k)1887 1255 y Fu(2)1960 1215 y Fm(@)2011 1230 y Fu(+)2071 1215 y Fm(F)2134 1230 y Fk(\000)p Fu(+)2270 1215 y Fl(\000)2402 1148 y Fp(3)p 2380 1192 94 4 v 2380 1284 a Fm(k)2434 1255 y Fu(2)2500 1215 y Fm(F)2563 1230 y Fk(\000)p Fu(+)2704 1215 y Fp(=)28 b Fm(J)2862 1230 y Fk(\000)361 1448 y Fl(r)444 1407 y Fh(N)511 1448 y Fm(F)574 1463 y Fh(N)7 b Fu(+)713 1448 y Fp(=)838 1381 y(1)p 816 1425 V 816 1517 a Fm(k)870 1488 y Fu(2)936 1448 y Fm(@)992 1407 y Fh(\027)1036 1448 y Fm(F)1099 1463 y Fh(\027)t Fu(+)1219 1448 y Fp(+)1330 1381 y(1)p 1327 1425 55 4 v 1327 1517 a Fm(k)1408 1448 y(@)1459 1463 y Fu(+)1518 1448 y Fm(F)1581 1463 y Fk(\000)p Fu(+)1723 1448 y Fp(=)28 b Fm(J)1881 1463 y Fu(+)1972 1448 y Fm(;)166 1829 y Fp(or,)k(in)h(terms)h(of)e(v)m (ector)h(p)s(oten)m(tial,)374 2243 y Fl(r)457 2202 y Fh(N)524 2243 y Fm(F)587 2258 y Fh(N)7 b(\026)713 2243 y Fp(=)838 2176 y(1)p 816 2220 94 4 v 816 2311 a Fm(k)870 2282 y Fu(2)936 2243 y Fc(\003)p Fm(A)1086 2258 y Fh(\026)1155 2243 y Fl(\000)1287 2176 y Fp(1)p 1265 2220 V 1265 2311 a Fm(k)1319 2282 y Fu(2)1385 2243 y Fm(@)1436 2258 y Fh(\026)1483 2243 y Fp(\()p Fm(@)28 b Fl(\001)22 b Fm(A)p Fp(\))g(+)1894 2176 y(2)p 1891 2220 55 4 v 1891 2311 a Fm(k)1972 2243 y(@)2023 2258 y Fk(\000)2082 2243 y Fm(@)2133 2258 y Fu(+)2193 2243 y Fm(A)2266 2258 y Fh(\026)2334 2243 y Fl(\000)2444 2176 y Fp(2)p Fm(\036)p 2444 2220 107 4 v 2450 2311 a(k)2504 2282 y Fu(2)2577 2243 y Fm(@)2628 2258 y Fu(+)2688 2243 y Fm(@)2739 2258 y Fu(+)2798 2243 y Fm(A)2871 2258 y Fh(\026)806 2481 y Fl(\000)896 2414 y Fp(1)p 893 2458 55 4 v 893 2549 a Fm(k)974 2481 y(@)1025 2496 y Fh(\026)1072 2481 y Fm(@)1123 2496 y Fu(+)1182 2481 y Fm(A)1255 2496 y Fk(\000)1337 2481 y Fl(\000)1449 2414 y Fp(1)p 1446 2458 V 1446 2549 a Fm(k)1527 2481 y(@)1578 2496 y Fh(\026)1625 2481 y Fm(@)1676 2496 y Fk(\000)1736 2481 y Fm(A)1809 2496 y Fu(+)1890 2481 y Fp(+)1998 2414 y(2)p Fm(\036)p 1998 2458 107 4 v 2004 2549 a(k)2058 2521 y Fu(2)2131 2481 y Fm(@)2182 2496 y Fh(\026)2229 2481 y Fm(@)2280 2496 y Fu(+)2340 2481 y Fm(A)2413 2496 y Fu(+)2499 2481 y Fp(=)28 b Fm(J)2657 2496 y Fh(\026)361 2784 y Fl(r)444 2743 y Fh(N)511 2784 y Fm(F)574 2799 y Fh(N)7 b Fk(\000)713 2784 y Fp(=)838 2716 y(1)p 816 2760 94 4 v 816 2852 a Fm(k)870 2823 y Fu(2)936 2784 y Fc(\003)p Fm(A)1086 2799 y Fk(\000)1168 2784 y Fl(\000)1300 2716 y Fp(1)p 1277 2760 V 1277 2852 a Fm(k)1331 2823 y Fu(2)1397 2784 y Fm(@)1448 2799 y Fk(\000)1508 2784 y Fp(\()p Fm(@)28 b Fl(\001)22 b Fm(A)p Fp(\))g(+)1918 2716 y(1)p 1916 2760 55 4 v 1916 2852 a Fm(k)1996 2784 y(@)2047 2799 y Fk(\000)2107 2784 y Fm(@)2158 2799 y Fu(+)2217 2784 y Fm(A)2290 2799 y Fk(\000)2372 2784 y Fl(\000)2481 2716 y Fp(2)p Fm(\036)p 2481 2760 107 4 v 2487 2852 a(k)2541 2823 y Fu(2)2614 2784 y Fm(@)2665 2799 y Fu(+)2725 2784 y Fm(@)2776 2799 y Fu(+)2836 2784 y Fm(A)2909 2799 y Fk(\000)806 3022 y Fl(\000)896 2954 y Fp(1)p 893 2999 55 4 v 893 3090 a Fm(k)974 3022 y(@)1025 3037 y Fk(\000)1084 3022 y Fm(@)1135 3037 y Fk(\000)1195 3022 y Fm(A)1268 3037 y Fu(+)1349 3022 y Fp(+)1457 2954 y(2)p Fm(\036)p 1457 2999 107 4 v 1463 3090 a(k)1517 3061 y Fu(2)1590 3022 y Fm(@)1641 3037 y Fk(\000)1701 3022 y Fm(@)1752 3037 y Fu(+)1811 3022 y Fm(A)1884 3037 y Fu(+)1966 3022 y Fp(+)2096 2954 y(3)p 2074 2999 94 4 v 2074 3090 a Fm(k)2128 3061 y Fu(2)2194 3022 y Fm(@)2245 3037 y Fu(+)2304 3022 y Fm(A)2377 3037 y Fk(\000)2459 3022 y Fl(\000)2591 2954 y Fp(3)p 2568 2999 V 2568 3090 a Fm(k)2622 3061 y Fu(2)2688 3022 y Fm(@)2739 3037 y Fk(\000)2799 3022 y Fm(A)2872 3037 y Fu(+)2959 3022 y Fp(=)27 b Fm(J)3116 3037 y Fk(\000)361 3319 y Fl(r)444 3278 y Fh(N)511 3319 y Fm(F)574 3334 y Fh(N)7 b Fu(+)713 3319 y Fp(=)838 3252 y(1)p 816 3296 V 816 3388 a Fm(k)870 3359 y Fu(2)936 3319 y Fc(\003)p Fm(A)1086 3334 y Fu(+)1168 3319 y Fl(\000)1300 3252 y Fp(1)p 1277 3296 V 1277 3388 a Fm(k)1331 3359 y Fu(2)1397 3319 y Fm(@)1448 3334 y Fu(+)1508 3319 y Fp(\()p Fm(@)28 b Fl(\001)22 b Fm(A)p Fp(\))g Fl(\000)1920 3252 y Fp(1)p 1917 3296 55 4 v 1917 3388 a Fm(k)1998 3319 y(@)2049 3334 y Fu(+)2109 3319 y Fm(@)2160 3334 y Fu(+)2219 3319 y Fm(A)2292 3334 y Fk(\000)2373 3319 y Fp(+)2484 3252 y(1)p 2481 3296 V 2481 3388 a Fm(k)2562 3319 y(@)2613 3334 y Fk(\000)2673 3319 y Fm(@)2724 3334 y Fu(+)2783 3319 y Fm(A)2856 3334 y Fu(+)2943 3319 y Fp(=)27 b Fm(J)3100 3334 y Fu(+)3192 3319 y Fm(;)166 3593 y Fp(where)34 b Fc(\003)p Fm(A)598 3608 y Fh(\026)673 3593 y Fp(:=)27 b Fm(\021)855 3557 y Fh(\032\027)934 3593 y Fm(@)985 3608 y Fh(\032)1026 3593 y Fm(@)1077 3608 y Fh(\027)1121 3593 y Fm(A)1194 3608 y Fh(\026)1241 3593 y Fp(,)32 b Fm(@)c Fl(\001)22 b Fm(A)28 b Fp(:=)g Fm(\021)1713 3557 y Fh(\026\027)1798 3593 y Fm(@)1849 3608 y Fh(\026)1896 3593 y Fm(A)1969 3608 y Fh(\027)2012 3593 y Fp(.)166 3827 y(Ob)m(viously)-8 b(,)34 b(the)f(6DME)f(are)g Fm(O)1359 3842 y Fu(0)1398 3827 y Fp(\(2)p Fm(;)17 b Fp(4\))o(-in)m(v)-5 b(arian)m(t.)32 b(They)i(are)e(also)g Fi(R)p Fp(-in)m(v)-5 b(arian)m(t)32 b(in)g(the)166 3947 y(canonical)42 b(case)h(when)f Fl(A)g Fp(and)f Fl(J)60 b Fp(c)m(hange)42 b(resp)s(ectiv)m(ely)j(according)d (to)f(the)h(actions)166 4068 y Fm(T)237 4031 y Fu(\(1\))364 4068 y Fp(and)32 b Fm(T)624 4031 y Fu(\(3\))751 4068 y Fp(of)g Fi(R)p Fp(.)166 4302 y(W)-8 b(e)43 b(p)s(erform)f(the)h Fi(R)p Fp(-reduction)f(of)g(6DME,)g(follo)m(wing)h(the)g(general)f(pro) s(cedure)i(de-)166 4422 y(scrib)s(ed)c(in)g(Sec.)g(5)f(in)g(P)m(art)g (I.)79 b(F)-8 b(rom)38 b(the)i(v)-5 b(anishing)39 b(of)g(the)h(Lie)f (deriv)-5 b(ativ)m(e)41 b(of)d(the)166 4542 y(action)d(\(8\))g(of)g Fi(R)g Fp(on)g(the)h Fi(R)p Fp(-in)m(v)-5 b(arian)m(t)35 b(di\013eren)m(tial)h(1-forms)f Fl(A)d(2)h Fm(C)2798 4506 y Fk(1)2872 4542 y Fp(\()p Fm(\034)2963 4506 y Fk(\003)3003 4542 y Fp(\()p Fi(R)3113 4506 y Fu(6)3152 4542 y Fp(\)\))3228 4557 y Ff(R)3280 4542 y Fp(,)j(w)m(e)166 4663 y(obtain)1363 4836 y Fm(\030)1411 4794 y Fh(m)1504 4768 y Fm(@)5 b Fl(A)1640 4783 y Fh(n)p 1504 4812 184 4 v 1510 4904 a Fm(@)g(\030)1614 4875 y Fh(m)1719 4836 y Fp(+)22 b Fm(\025)17 b Fl(A)1971 4851 y Fh(n)2045 4836 y Fp(=)27 b(0)33 b Fm(:)166 5058 y Fp(In)g(the)g(case)g Fm(\025)28 b Fp(=)g(1)k(this)h (equation)g(reads)651 5345 y Fm(@)702 5360 y Fk(\000)762 5345 y Fm(A)835 5360 y Fh(\026;)11 b Fu(+)967 5345 y Fp(\()p Fm(x;)17 b(k)s(;)g(\036)p Fp(\))28 b(=)f(0)195 b Fm(@)1724 5360 y Fk(\000)1784 5345 y Fm(A)1857 5360 y Fk(\000)1916 5345 y Fp(\()p Fm(x;)17 b(k)s(;)g(\036)p Fp(\))27 b(=)2390 5277 y(1)p 2388 5321 55 4 v 2388 5413 a Fm(k)2468 5345 y(A)2541 5360 y Fk(\000)2600 5345 y Fp(\()p Fm(x;)17 b(k)s(;)g(\036)p Fp(\))32 b Fm(:)290 b Fp(\(14\))1769 5712 y(9)p eop %%Page: 10 10 10 9 bop 166 83 a Fp(W)-8 b(e)28 b(restrict)g(the)f(\014rst)h (prolongation)f(of)f(\(14\))h(to)g(the)g(submanifold)h Fm(U)38 b Fp(\(2\),)27 b(transv)m(ersal)166 203 y(to)32 b(the)h Fi(R)p Fp(-orbits)g(in)f Fi(R)1020 167 y Fu(6)1060 203 y Fp(,)g(and)h(obtain)g(the)g(splitting)g(relations)472 508 y Fm(@)523 523 y Fk(\000)582 508 y Fm(A)655 523 y Fh(\026;)12 b Fu(+)788 508 y Fp(\()p Fm(x;)17 b(\036)p Fp(\))28 b(=)f(0)356 b Fm(@)1608 523 y Fk(\000)1667 508 y Fm(A)1740 523 y Fk(\000)1799 508 y Fp(\()p Fm(x;)17 b(\036)p Fp(\))28 b(=)f Fl(\000)p Fm(A)2313 523 y Fk(\000)2373 508 y Fp(\()p Fm(x;)17 b(\036)p Fp(\))361 644 y Fm(@)412 659 y Fk(\000)472 644 y Fm(@)523 659 y Fk(\000)582 644 y Fm(A)655 659 y Fh(\026;)12 b Fu(+)788 644 y Fp(\()p Fm(x;)17 b(\036)p Fp(\))28 b(=)f(0)228 b Fm(@)1480 659 y Fk(\000)1540 644 y Fm(@)1591 659 y Fk(\000)1650 644 y Fm(A)1723 659 y Fk(\000)1782 644 y Fp(\()p Fm(x;)17 b(\036)p Fp(\))28 b(=)f(0)33 b Fm(;)166 852 y Fp(where)h Fm(A)521 867 y Fh(\026;)p Fk(\000)p Fh(;)p Fu(+)717 852 y Fp(\()p Fm(x;)17 b(\036)p Fp(\))27 b(:=)h Fm(A)1181 867 y Fh(\026;)p Fk(\000)p Fh(;)p Fu(+)1376 852 y Fp(\()p Fm(x;)17 b Fp(1)p Fm(;)g(\036)p Fp(\).)166 1072 y(With)33 b(these)h(relations,)f(the)g Fi(R)p Fp(-reduced)h(6DME)e(read)394 1377 y Fc(\003)p Fm(A)544 1392 y Fh(\026)614 1377 y Fl(\000)22 b Fm(@)764 1392 y Fh(\026)811 1377 y Fp(\()p Fm(@)28 b Fl(\001)22 b Fm(A)p Fp(\))g Fl(\000)h Fp(2)p Fm(\036)17 b(@)1386 1392 y Fu(+)1445 1377 y Fm(@)1496 1392 y Fu(+)1555 1377 y Fm(A)1628 1392 y Fh(\026)1697 1377 y Fl(\000)23 b Fm(@)1848 1392 y Fh(\026)1895 1377 y Fm(@)1946 1392 y Fu(+)2005 1377 y Fm(A)2078 1392 y Fk(\000)2160 1377 y Fp(+)f(2)p Fm(\036)17 b(@)2433 1392 y Fh(\026)2479 1377 y Fm(@)2530 1392 y Fu(+)2589 1377 y Fm(A)2662 1392 y Fu(+)2749 1377 y Fp(=)28 b Fm(J)2907 1392 y Fh(\026)394 1513 y Fc(\003)p Fm(A)544 1528 y Fk(\000)626 1513 y Fl(\000)23 b Fp(2)p Fm(\036)17 b(@)901 1528 y Fu(+)959 1513 y Fm(@)1010 1528 y Fu(+)1070 1513 y Fm(A)1143 1528 y Fk(\000)1224 1513 y Fp(+)22 b(2)17 b Fm(@)1439 1528 y Fu(+)1498 1513 y Fm(A)1571 1528 y Fk(\000)1658 1513 y Fp(=)28 b Fm(J)1816 1528 y Fk(\000)3280 1513 y Fp(\(15\))394 1645 y Fc(\003)p Fm(A)544 1660 y Fu(+)626 1645 y Fl(\000)23 b Fm(@)777 1660 y Fu(+)836 1645 y Fp(\()p Fm(@)28 b Fl(\001)22 b Fm(A)p Fp(\))g Fl(\000)h Fm(@)1287 1660 y Fu(+)1346 1645 y Fm(@)1397 1660 y Fu(+)1457 1645 y Fm(A)1530 1660 y Fk(\000)1617 1645 y Fp(=)k Fm(J)1774 1660 y Fu(+)1866 1645 y Fm(:)166 1954 y Fp(The)46 b(restriction)g(to)f(the)h (submanifold)g Fm(M)56 b Fp(is)46 b(complicated)g(b)s(ecause)h(after)e (setting)166 2074 y Fm(\036)27 b Fp(=)h(0)k(in)h(\(15\))f(w)m(e)i (obtain)e(the)h(equations)394 2379 y Fc(\003)p Fm(A)544 2394 y Fh(\026)614 2379 y Fl(\000)22 b Fm(@)764 2394 y Fh(\026)811 2379 y Fp(\()p Fm(@)28 b Fl(\001)22 b Fm(A)p Fp(\))g Fl(\000)h Fm(@)1262 2394 y Fh(\026)1309 2379 y Fm(@)1360 2394 y Fu(+)1420 2379 y Fm(A)1493 2394 y Fk(\000)1580 2379 y Fp(=)k Fm(J)1737 2394 y Fh(\026)394 2514 y Fc(\003)p Fm(A)544 2529 y Fk(\000)626 2514 y Fp(+)22 b(2)17 b Fm(@)841 2529 y Fu(+)900 2514 y Fm(A)973 2529 y Fk(\000)1060 2514 y Fp(=)27 b Fm(J)1217 2529 y Fk(\000)3280 2514 y Fp(\(16\))394 2647 y Fc(\003)p Fm(A)544 2662 y Fu(+)626 2647 y Fl(\000)c Fm(@)777 2662 y Fu(+)836 2647 y Fp(\()p Fm(@)28 b Fl(\001)22 b Fm(A)p Fp(\))g Fl(\000)h Fm(@)1287 2662 y Fu(+)1346 2647 y Fm(@)1397 2662 y Fu(+)1457 2647 y Fm(A)1530 2662 y Fk(\000)1617 2647 y Fp(=)k Fm(J)1774 2662 y Fu(+)166 2856 y Fp(whic)m(h)f(con)m(tain)f Fm(@)824 2871 y Fu(+)907 2856 y Fp(deriv)-5 b(ativ)m(es,)27 b(i.e.,)e(the)f Fi(R)p Fp(-reduced)i(6DME)e(are)g(not)g(in)m(ternal)h(for)f Fm(M)10 b Fp(.)166 2976 y(T)-8 b(o)32 b(restrict)i(the)e(non-in)m (ternal)h(DO)f(in)g(\(16\))g(to)g Fm(M)10 b Fp(,)33 b(w)m(e)h(need)f (some)g(additional)f(infor-)166 3097 y(mation,)j(e.g.,)g(a)g(splitting) g(relation)g(for)f(the)h(SES)h(\(8\))e(in)h(P)m(art)g(I.)70 b(Besides,)37 b(w)m(e)f(w)m(an)m(t)166 3217 y(the)d(restricted)h(DO)d (to)h(b)s(e)h(conformally)g(in)m(v)-5 b(arian)m(t.)33 b(Splitting)g(relations)g(of)f(this)h(kind)166 3338 y(can)40 b(b)s(e)g(obtained)g(b)m(y)g(considering)h(the)f(k)m(ernel)h(of)e(a)g (manifestly)i Fm(O)2788 3353 y Fu(0)2827 3338 y Fp(\(2)p Fm(;)17 b Fp(4\)-in)m(v)-5 b(arian)m(t)166 3458 y(DO)32 b(after)g(its)h Fi(R)p Fp(-reduction)g(and)g(restriction)h(to)e Fm(M)10 b Fp(.)166 3678 y(Let)28 b(us)h(c)m(ho)s(ose)g(as)f(a)g (splitting)h(relation)f(the)h(manifestly)h Fm(O)2395 3693 y Fu(0)2434 3678 y Fp(\(2)p Fm(;)17 b Fp(4\))o(-in)m(v)-5 b(arian)m(t)28 b(equation)1324 3936 y Fl(r)1407 3951 y Fu(6)1469 3936 y Fl(\001)21 b(A)28 b Fp(=)1794 3869 y Fm(@)p 1739 3913 167 4 v 1739 4005 a(@)5 b(\030)1838 4020 y Fh(m)1932 3936 y Fl(A)2012 3951 y Fh(m)2106 3936 y Fp(=)27 b(0)32 b Fm(:)963 b Fp(\(17\))166 4290 y(Its)33 b Fi(R)p Fp(-reduction)g(reads)559 4498 y Fm(@)28 b Fl(\001)22 b Fm(A)p Fp(\()p Fm(x;)17 b(\036)p Fp(\))22 b(+)g Fm(@)1165 4513 y Fu(+)1224 4498 y Fm(A)1297 4513 y Fk(\000)1356 4498 y Fp(\()p Fm(x;)17 b(\036)p Fp(\))22 b Fl(\000)h Fp(2)p Fm(\036)17 b(@)1886 4513 y Fu(+)1944 4498 y Fm(A)2017 4513 y Fu(+)2077 4498 y Fp(\()p Fm(x;)g(\036)p Fp(\))k(+)h(2)17 b Fm(A)2568 4513 y Fu(+)2627 4498 y Fp(\()p Fm(x:\036)p Fp(\))28 b(=)f(0)33 b Fm(:)197 b Fp(\(18\))166 4807 y(The)34 b(\014rst)f(prolongation)f(of)g(\(18\),)g(restricted)i(to)e Fm(M)10 b Fp(,)34 b(yields)394 5111 y Fm(@)28 b Fl(\001)22 b Fm(A)g Fp(+)g Fm(@)767 5126 y Fu(+)827 5111 y Fm(A)900 5126 y Fk(\000)981 5111 y Fp(+)g(2)p Fm(A)1201 5126 y Fu(+)1288 5111 y Fp(=)28 b(0)394 5244 y Fm(@)445 5259 y Fh(\026)492 5244 y Fp(\()p Fm(@)g Fl(\001)22 b Fm(A)p Fp(\))g(+)g Fm(@)941 5259 y Fh(\026)989 5244 y Fm(@)1040 5259 y Fu(+)1099 5244 y Fm(A)1172 5259 y Fk(\000)1253 5244 y Fp(+)g(2)17 b Fm(@)1468 5259 y Fh(\026)1515 5244 y Fm(A)1588 5259 y Fu(+)1675 5244 y Fp(=)27 b(0)1453 b(\(19\))394 5380 y Fm(@)445 5395 y Fu(+)505 5380 y Fp(\()p Fm(@)28 b Fl(\001)22 b Fm(A)p Fp(\))g(+)g Fm(@)954 5395 y Fu(+)1013 5380 y Fm(@)1064 5395 y Fu(+)1124 5380 y Fm(A)1197 5395 y Fk(\000)1284 5380 y Fp(=)27 b(0)33 b Fm(:)1745 5712 y Fp(10)p eop %%Page: 11 11 11 10 bop 166 83 a Fp(This)37 b(is)f(not)g(a)f(splitting)i(of)e(the)h (SES)h(\(8\))e(in)h(P)m(art)g(I,)g(but)g(pro)m(vides)i(us)e(with)h(a)e (su\016-)166 203 y(cien)m(t)e(set)g(of)f(splitting)h(relations)f(for)g (it.)g(Com)m(bining)h(\(16\))f(with)h(\(19\),)e(w)m(e)i(obtain)f(the) 166 324 y(equations)394 640 y Fc(\003)p Fm(A)544 655 y Fh(\026)592 640 y Fp(\()p Fm(x)p Fp(\))22 b(+)g(2)17 b Fm(@)960 655 y Fh(\026)1007 640 y Fm(A)1080 655 y Fu(+)1139 640 y Fp(\()p Fm(x)p Fp(\))28 b(=)f Fm(J)1455 655 y Fh(\026)1502 640 y Fp(\()p Fm(x)p Fp(\))394 776 y Fc(\003)p Fm(A)544 791 y Fk(\000)604 776 y Fp(\()p Fm(x)p Fp(\))22 b Fl(\000)h Fp(2)17 b Fm(@)28 b Fl(\001)21 b Fm(A)p Fp(\()p Fm(x)p Fp(\))i Fl(\000)g Fp(4)17 b Fm(A)1517 791 y Fu(+)1575 776 y Fp(\()p Fm(x)p Fp(\))28 b(=)g Fm(J)1892 791 y Fk(\000)1951 776 y Fp(\()p Fm(x)p Fp(\))1198 b(\(20\))394 909 y Fc(\003)p Fm(A)544 924 y Fu(+)604 909 y Fp(\()p Fm(x)p Fp(\))28 b(=)g Fm(J)921 924 y Fu(+)979 909 y Fp(\()p Fm(x)p Fp(\))33 b Fm(;)166 1130 y Fp(often)g(referred)g(to)f(as)h Fj(c)-5 b(onformal)34 b(ele)-5 b(ctr)g(o)g(dynamics)33 b(e)-5 b(quations)40 b Fp([7,5].)166 1350 y(Numerous)34 b(mo)s(di\014cations)g (of)e(\(20\),)h(discussed)i(in)e(the)g(literature,)h(can)f(b)s(e)g (easily)h(ob-)166 1471 y(tained)f(from)f(them.)i(F)-8 b(or)32 b(example,)i(eliminating)f Fm(A)2137 1486 y Fu(+)2196 1471 y Fp(\()p Fm(x)p Fp(\),)g(w)m(e)h(obtain)420 1844 y Fm(@)476 1803 y Fh(\026)524 1844 y Fm(F)587 1859 y Fh(\026\027)672 1844 y Fp(\()p Fm(x)p Fp(\))23 b(+)934 1777 y(1)p 934 1821 49 4 v 934 1913 a(2)1009 1844 y Fm(@)1060 1859 y Fh(\027)1104 1844 y Fc(\003)p Fm(A)1254 1859 y Fk(\000)1313 1844 y Fp(\()p Fm(x)p Fp(\))17 b(=)g Fm(J)1608 1859 y Fh(\027)1651 1844 y Fp(\()p Fm(x)p Fp(\))22 b(+)1912 1777 y(1)p 1912 1821 V 1912 1913 a(2)1988 1844 y Fm(@)2039 1859 y Fh(\027)2082 1844 y Fm(J)2136 1859 y Fk(\000)2195 1844 y Fp(\()p Fm(x)p Fp(\))3280 1962 y(\(21\))371 2025 y(1)p 371 2069 V 371 2161 a(4)446 2092 y Fc(\003)523 2051 y Fu(2)563 2092 y Fm(A)636 2107 y Fk(\000)696 2092 y Fp(\()p Fm(x)p Fp(\))g Fl(\000)959 2025 y Fp(1)p 959 2069 V 959 2161 a(2)1034 2092 y Fc(\003)p Fm(@)28 b Fl(\001)22 b Fm(A)p Fp(\()p Fm(x)p Fp(\))17 b(=)g Fm(J)1608 2107 y Fu(+)1667 2092 y Fp(\()p Fm(x)p Fp(\))22 b Fl(\000)1930 2025 y Fp(1)p 1930 2069 V 1930 2161 a(4)2005 2092 y Fc(\003)p Fm(J)2136 2107 y Fk(\000)2196 2092 y Fp(\()p Fm(x)p Fp(\))32 b Fm(:)166 2444 y Fp(Some)41 b(of)e(the)i(conformally)g(in)m(v)-5 b(arian)m(t)40 b(conditions)i(\(12\))d(can)h(b)s(e)h(imp)s(osed)g (indep)s(en-)166 2564 y(den)m(tly)36 b(on)d Fl(A)h Fp(and)g Fl(J)18 b Fp(.)34 b(F)-8 b(or)33 b(example,)j(setting)e Fm(J)2016 2579 y Fk(\000)2075 2564 y Fp(\()p Fm(x)p Fp(\))d(=)e(0)34 b(in)g(\(21\),)g(w)m(e)h(arriv)m(e)g(at)e(the)166 2685 y(system)i([7,22,11])420 3058 y Fm(@)476 3017 y Fh(\026)524 3058 y Fm(F)587 3073 y Fh(\026\027)672 3058 y Fp(\()p Fm(x)p Fp(\))23 b(+)934 2991 y(1)p 934 3035 V 934 3127 a(2)1009 3058 y Fm(@)1060 3073 y Fh(\027)1104 3058 y Fc(\003)p Fm(A)1254 3073 y Fk(\000)1313 3058 y Fp(\()p Fm(x)p Fp(\))17 b(=)g Fm(J)1608 3073 y Fh(\027)1651 3058 y Fp(\()p Fm(x)p Fp(\))371 3224 y(1)p 371 3268 V 371 3360 a(4)446 3292 y Fc(\003)523 3250 y Fu(2)563 3292 y Fm(A)636 3307 y Fk(\000)696 3292 y Fp(\()p Fm(x)p Fp(\))22 b Fl(\000)959 3224 y Fp(1)p 959 3268 V 959 3360 a(2)1034 3292 y Fc(\003)p Fm(@)28 b Fl(\001)22 b Fm(A)p Fp(\()p Fm(x)p Fp(\))17 b(=)g Fm(J)1608 3307 y Fu(+)1667 3292 y Fp(\()p Fm(x)p Fp(\))33 b Fm(:)166 3543 y Fp(If,)g(in)f(addition,)h Fm(A)889 3558 y Fk(\000)949 3543 y Fp(\()p Fm(x)p Fp(\))27 b(=)h(0,)k(w)m(e)i(obtain)f(the)g(equations)g([18,5])551 3860 y Fm(@)607 3819 y Fh(\026)655 3860 y Fm(F)718 3875 y Fh(\026\027)803 3860 y Fp(\()p Fm(x)p Fp(\))17 b(=)g Fm(J)1098 3875 y Fh(\027)1141 3860 y Fp(\()p Fm(x)p Fp(\))361 4053 y Fl(\000)448 3985 y Fp(1)p 448 4029 V 448 4121 a(2)524 4053 y Fc(\003)p Fm(@)28 b Fl(\001)22 b Fm(A)p Fp(\()p Fm(x)p Fp(\))17 b(=)g Fm(J)1098 4068 y Fu(+)1157 4053 y Fp(\()p Fm(x)p Fp(\))33 b Fm(:)166 4482 y Fq(5)112 b(Conformally)39 b(In)m(v)-6 b(arian)m(t)38 b(Gauge)g(T)-9 b(ransformations)166 4822 y Fp(In)36 b(this)h(section,)g(w)m(e)g(deriv) m(e)g(the)f(most)h(general)f(gauge)g(transformations)g(preserving)166 4943 y(the)e(conformal)f(in)m(v)-5 b(ariance.)34 b(Let)g Fl(A)1532 4907 y Fk(0)1583 4943 y Fp(=)29 b Fl(A)22 b Fp(+)g(d)p Fl(S)41 b Fp(b)s(e)34 b(a)f(gauge)g(transformation)g(in)g Fi(R)3387 4907 y Fu(6)3427 4943 y Fp(.)166 5063 y(The)h Fi(R)p Fp(-in)m(v)-5 b(ariance)33 b(of)f Fl(A)g Fp(and)h Fl(A)1425 5027 y Fk(0)1480 5063 y Fp(imp)s(oses)h(the)f(conditions)361 5380 y Fm(@)412 5395 y Fk(\000)472 5380 y Fm(@)523 5395 y Fh(\026;)12 b Fu(+)656 5380 y Fm(S)6 b Fp(\()p Fm(x;)17 b(k)s(;)g(\036)p Fp(\))f(=)g(0)1745 5712 y(11)p eop %%Page: 12 12 12 11 bop 435 132 a Fm(@)486 147 y Fk(\000)545 132 y Fm(@)596 147 y Fk(\000)656 132 y Fm(S)6 b Fp(\()p Fm(x;)17 b(k)s(;)g(\036)p Fp(\))f(=)g Fl(\000)1268 64 y Fp(1)p 1265 108 55 4 v 1265 200 a Fm(k)1346 132 y(@)1397 147 y Fk(\000)1457 132 y Fm(S)6 b Fp(\()p Fm(x;)17 b(k)s(;)g(\036)p Fp(\))166 396 y(on)33 b Fl(S)40 b Fp(\(cf.)33 b(\(14\)\).)f(Their)h (solution)g(is)1198 629 y Fm(S)6 b Fp(\()p Fm(x;)17 b(k)s(;)g(\036)p Fp(\))27 b(=)h Fm(s)17 b Fp(ln)f Fl(j)p Fm(k)s Fl(j)22 b Fp(+)g Fm(t)p Fp(\()p Fm(x;)17 b(\036)p Fp(\))32 b Fm(;)837 b Fp(\(22\))166 968 y(where)25 b Fm(s)e Fp(is)h(a)g(constan)m (t)g(and)g Fm(t)f Fp(is)i(an)e(arbitrary)h(function)g(of)f Fm(x)h Fp(and)f Fm(\036)p Fp(.)h(The)g(dimensional)166 1088 y(reduction)34 b(of)e(\(22\))g(yields)472 1423 y Fm(S)6 b Fp(\()p Fm(x;)17 b(\036)p Fp(\))f(:=)g Fm(S)6 b Fp(\()p Fm(x;)17 b Fp(1)p Fm(;)g(\036)p Fp(\))27 b(=)h Fm(t)p Fp(\()p Fm(x;)17 b(\036)p Fp(\))32 b Fm(;)3280 1508 y Fp(\(23\))361 1594 y Fm(@)412 1609 y Fk(\000)472 1594 y Fm(S)6 b Fp(\()p Fm(x;)17 b(\036)p Fp(\))f(:=)g Fm(@)957 1609 y Fk(\000)1017 1594 y Fm(S)6 b Fp(\()p Fm(x;)17 b(k)s(;)g(\036)p Fp(\))o Fl(j)1441 1609 y Fh(k)r Fu(=1)1601 1594 y Fp(=)28 b Fm(s)k(:)166 1933 y Fp(The)i(actions)f(of)f Fm(K)7 b Fp(\()p Fm(c)p Fp(\))32 b(and)h Fm(D)s Fp(\()p Fm(d)p Fp(\))f(on)g Fm(S)6 b Fp(\()p Fm(x;)17 b(k)s(;)g(\036)p Fp(\))32 b(\(22\))g(are)361 2281 y Fm(D)s Fp(\()p Fm(d)p Fp(\)\()p Fm(S)6 b Fp(\)\()p Fm(x;)17 b(k)s(;)g(\036)p Fp(\))e(=)i Fm(s)g(d)k Fp(+)h Fm(S)6 b Fp(\()p Fm(e)1535 2239 y Fk(\000)p Fh(d)1630 2281 y Fm(x;)17 b(k)s(;)g(e)1872 2239 y Fk(\000)p Fu(2)p Fh(d)2003 2281 y Fm(\036)p Fp(\))363 2417 y Fm(K)7 b Fp(\()p Fm(c)p Fp(\)\()p Fm(S)f Fp(\)\()p Fm(x;)17 b(k)s(;)g(\036)p Fp(\))f(=)h Fm(s)g Fp(ln)f Fl(j)p Fm(p)p Fp(\()p Fl(\000)p Fm(c;)h(x;)g(\036)p Fp(\))p Fl(j)k Fp(+)h Fm(S)6 b Fp(\()2037 2376 y Fk(0)2061 2417 y Fm(x;)17 b(k)s(;)2241 2376 y Fk(0)2281 2417 y Fm(\036)p Fp(\))32 b Fm(;)166 2650 y Fp(where)458 2614 y Fk(0)481 2650 y Fm(x)536 2614 y Fh(\026)627 2650 y Fp(:=)45 b Fm(p)824 2614 y Fk(\000)p Fu(1)918 2650 y Fp(\()p Fl(\000)p Fm(c;)17 b(x;)g(\036)p Fp(\)[)p Fm(x)1396 2614 y Fh(\026)1472 2650 y Fp(+)28 b Fm(c)1618 2614 y Fh(\026)1665 2650 y Fp(\()p Fm(x)1758 2614 y Fu(2)1827 2650 y Fl(\000)h Fp(2)p Fm(\036)p Fp(\)],)2174 2614 y Fk(0)2197 2650 y Fm(\036)44 b Fp(:=)h Fm(p)2496 2614 y Fk(\000)p Fu(2)2590 2650 y Fp(\()p Fl(\000)p Fm(c;)17 b(x;)g(\036)p Fp(\))g Fm(\036)p Fp(.)42 b(The)h(in-)166 2770 y(\014nitesimal)34 b(form)e(of)h(these)g (relations)g(is)361 3105 y Fm(\016)t(S)6 b Fp(\()p Fm(x;)17 b(k)s(;)g(\036)p Fp(\))f(=)g Fm(d)h Fp([)p Fm(s)22 b Fl(\000)h Fp(\()p Fm(x)f Fl(\001)g Fm(@)28 b Fp(+)22 b(2)p Fm(\036)17 b(@)1693 3120 y Fu(+)1752 3105 y Fp(\))p Fm(S)6 b Fp(\()p Fm(x;)17 b(k)s(;)g(\036)p Fp(\)])361 3263 y Fm(\016)t(S)6 b Fp(\()p Fm(x;)17 b(k)s(;)g(\036)p Fp(\))f(=)g Fm(c)955 3222 y Fh(\026)1019 3166 y Fg(n)1074 3263 y Fp(2)p Fm(sx)1224 3278 y Fh(\026)1293 3263 y Fp(+)1391 3166 y Fg(h)1430 3263 y Fp(\()p Fm(x)1523 3222 y Fu(2)1585 3263 y Fl(\000)23 b Fp(2)p Fm(\036)p Fp(\))p Fm(@)1881 3278 y Fh(\026)1949 3263 y Fl(\000)g Fp(2)p Fm(x)2153 3278 y Fh(\026)2200 3263 y Fm(x)f Fl(\001)g Fm(@)28 b Fl(\000)23 b Fp(4)p Fm(\036)17 b(x)2685 3278 y Fh(\026)2731 3263 y Fm(@)2782 3278 y Fu(+)2842 3166 y Fg(i)2897 3263 y Fm(S)6 b Fp(\()p Fm(x;)17 b(k)s(;)g(\036)p Fp(\))3294 3166 y Fg(o)3398 3263 y Fm(:)166 3602 y Fp(After)33 b(the)g Fi(R)p Fp(-reduction)g(and)f(restriction)i(to)e Fm(M)44 b Fp(w)m(e)33 b(obtain)361 3937 y Fm(D)s Fp(\()p Fm(d)p Fp(\)\()p Fm(S)6 b Fp(\)\()p Fm(x;)17 b(k)s(;)g(\036)p Fp(\))f(=)h Fm(s)g(d)22 b Fp(+)g Fm(S)6 b Fp(\()p Fm(e)1537 3896 y Fk(\000)p Fh(d)1632 3937 y Fm(x)p Fp(\))28 b Fl(\031)g Fm(S)6 b Fp(\()p Fm(x)p Fp(\))22 b(+)g Fm(d)17 b Fp([)p Fm(s)22 b Fl(\000)h Fm(x)f Fl(\001)g Fm(@)g(S)6 b Fp(\()p Fm(x)p Fp(\)])363 4073 y Fm(K)h Fp(\()p Fm(c)p Fp(\)\()p Fm(S)f Fp(\)\()p Fm(x;)17 b(k)s(;)g(\036)p Fp(\))g(=)g Fm(s)g Fp(ln)g Fl(j)p Fm(p)p Fp(\()p Fl(\000)p Fm(c;)g(x)p Fp(\))p Fl(j)22 b Fp(+)g Fm(S)6 b Fp(\()1938 4032 y Fk(0)1961 4073 y Fm(x)p Fp(\))1060 4231 y Fl(\031)17 b Fm(S)6 b Fp(\()p Fm(x)p Fp(\))23 b(+)f Fm(c)1514 4189 y Fh(\026)1577 4134 y Fg(h)1616 4231 y Fp(2)p Fm(sx)1766 4246 y Fh(\026)1835 4231 y Fp(+)g(\()p Fm(x)2026 4189 y Fu(2)2066 4231 y Fm(@)2117 4246 y Fh(\026)2186 4231 y Fl(\000)h Fp(2)p Fm(x)2390 4246 y Fh(\026)2436 4231 y Fm(x)g Fl(\001)f Fm(@)5 b Fp(\))p Fm(S)h Fp(\()p Fm(x)p Fp(\))2855 4134 y Fg(i)2944 4231 y Fm(;)166 4480 y Fp(where)34 b Fm(S)6 b Fp(\()p Fm(x)p Fp(\))28 b(:=)f Fm(S)6 b Fp(\()p Fm(x;)17 b Fp(0\),)1152 4444 y Fk(0)1176 4480 y Fm(x)1231 4444 y Fh(\026)1305 4480 y Fp(:=)28 b Fm(p)p Fp(\()p Fl(\000)p Fm(c;)17 b(x;)g Fp(0\)\()p Fm(x)1965 4444 y Fh(\026)2034 4480 y Fp(+)22 b Fm(c)2174 4444 y Fh(\026)2221 4480 y Fm(x)2276 4444 y Fu(2)2316 4480 y Fp(\).)166 4707 y(The)35 b(gauge)f(transformations)g(for)g(the)g(conformal)h(electro)s(dynamics) h(equations)f(\(20\))166 4827 y(m)m(ust)43 b(b)s(e)e(compatible)i(with) f(the)g(condition)g(\(17\))f(used)h(as)g(a)f(splitting)i(relation)e (for)166 4947 y(obtaining)35 b(\(20\).)g(Therefore,)i(the)f(gauge)f (function)h(m)m(ust)h(satisfy)f(the)g(6-dimensional)166 5068 y(\\w)m(a)m(v)m(e)e(equation")f Fc(\003)977 5083 y Fu(6)1017 5068 y Fl(S)j Fp(=)27 b(0,)33 b(whic)m(h)h(in)f Fm(Q)p Fp(-co)s(ordinates)f(reads)198 5277 y(1)p 176 5321 94 4 v 176 5413 a Fm(k)230 5384 y Fu(2)296 5345 y Fc(\003)p Fm(S)6 b Fp(\()p Fm(x;)17 b(k)s(;)g(\036)p Fp(\))h(+)896 5277 y(2)p 893 5321 55 4 v 893 5413 a Fm(k)974 5345 y(@)1025 5360 y Fk(\000)1084 5345 y Fm(@)1135 5360 y Fu(+)1195 5345 y Fm(S)6 b Fp(\()p Fm(x;)17 b(k)s(;)g(\036)p Fp(\))g Fl(\000)1715 5277 y Fp(2)p Fm(\036)p 1715 5321 107 4 v 1721 5413 a(k)1775 5384 y Fu(2)1849 5345 y Fm(@)1900 5360 y Fu(+)1959 5345 y Fm(@)2010 5360 y Fu(+)2070 5345 y Fm(S)6 b Fp(\()p Fm(x;)17 b(k)s(;)g(\036)p Fp(\))h(+)2611 5277 y(2)p 2589 5321 94 4 v 2589 5413 a Fm(k)2643 5384 y Fu(2)2709 5345 y Fm(@)2760 5360 y Fu(+)2820 5345 y Fm(S)6 b Fp(\()p Fm(x;)17 b(k)s(;)g(\036)p Fp(\))27 b(=)g(0)k Fm(:)1745 5712 y Fp(12)p eop %%Page: 13 13 13 12 bop 166 83 a Fp(After)33 b(the)g Fi(R)p Fp(-reduction,)g(w)m(e)g (obtain)g(for)f Fm(t)p Fp(\()p Fm(x;)17 b(\036)p Fp(\))32 b(the)h(condition)868 312 y Fc(\003)p Fm(t)p Fp(\()p Fm(x;)17 b(\036)p Fp(\))23 b Fl(\000)g Fp(2)p Fm(\036)17 b(@)1511 327 y Fu(+)1569 312 y Fm(@)1620 327 y Fu(+)1680 312 y Fm(t)p Fp(\()p Fm(x;)g(\036)p Fp(\))22 b(+)g(2)17 b Fm(@)2185 327 y Fu(+)2244 312 y Fm(t)p Fp(\()p Fm(x;)g(\036)p Fp(\))27 b(=)h(0)k Fm(;)166 541 y Fp(whose)i(\014rst)f(prolongation,)f (restricted)i(to)e Fm(M)10 b Fp(,)34 b(yields)g(the)f(relations)394 922 y Fm(@)445 937 y Fu(+)505 922 y Fm(t)p Fp(\()p Fm(x;)17 b Fp(0\))22 b(+)894 855 y(1)p 894 899 49 4 v 894 990 a(2)969 922 y Fc(\003)p Fm(t)p Fp(\()p Fm(x;)17 b Fp(0\))29 b(=)e(0)32 b Fm(;)1735 b Fp(\(24\))394 1098 y Fm(@)445 1113 y Fu(+)505 1098 y Fc(\003)p Fm(t)p Fp(\()p Fm(x;)17 b Fp(0\))28 b(=)g(0)k Fm(:)2199 b Fp(\(25\))166 1328 y(Acting)39 b(on)f(\(24\))f(with)i Fc(\003)g Fp(and)f(taking)g(in)m(to) g(accoun)m(t)h(\(25\),)f(w)m(e)h(obtain)f(the)h(in)m(ternal)166 1448 y(for)34 b Fm(M)46 b Fp(condition)35 b Fc(\003)964 1412 y Fu(2)1004 1448 y Fm(t)p Fp(\()p Fm(x)p Fp(\))c(=)g(0,)k(where)h Fm(t)p Fp(\()p Fm(x)p Fp(\))c(:=)f Fm(t)p Fp(\()p Fm(x;)17 b Fp(0\).)34 b(Due)h(to)f(\(23\))g(and)h(\(24\),)f(the)166 1568 y(conformally)f(in)m(v)-5 b(arian)m(t)33 b(\\gauge)f (transformations")h(in)g Fm(M)43 b Fp(are)374 1898 y Fm(A)447 1857 y Fk(0)447 1922 y Fh(\026)493 1898 y Fp(\()p Fm(x)p Fp(\))17 b(=)g Fm(A)807 1913 y Fh(\026)853 1898 y Fp(\()p Fm(x)p Fp(\))23 b(+)f Fm(@)1156 1913 y Fh(\026)1203 1898 y Fm(S)6 b Fp(\()p Fm(x)p Fp(\))361 2046 y Fm(A)434 2005 y Fk(0)434 2071 y(\000)493 2046 y Fp(\()p Fm(x)p Fp(\))17 b(=)g Fm(A)807 2061 y Fk(\000)866 2046 y Fp(\()p Fm(x)p Fp(\))22 b(+)g Fm(s)361 2243 y(A)434 2202 y Fk(0)434 2267 y Fu(+)493 2243 y Fp(\()p Fm(x)p Fp(\))17 b(=)g Fm(A)807 2258 y Fu(+)866 2243 y Fp(\()p Fm(x)p Fp(\))22 b Fl(\000)1129 2175 y Fp(1)p 1129 2220 V 1129 2311 a(2)1204 2243 y Fc(\003)p Fm(S)6 b Fp(\()p Fm(x)p Fp(\))33 b Fm(;)166 2503 y Fp(where)h(the)f(gauge)f(function)h Fm(S)6 b Fp(\()p Fm(x)p Fp(\))33 b(is)g(a)f(solution)h(to)1574 2732 y Fc(\003)1651 2691 y Fu(2)1691 2732 y Fm(S)6 b Fp(\()p Fm(x)p Fp(\))27 b(=)h(0)1212 b(\(26\))166 3066 y([7,11,5].)37 b(It)g(is)h(easy)g(to)f(c)m(hec)m(k)j(that)d(\(26\))g(is)h(su\016cien)m (t)h(for)e(the)h(gauge)f(in)m(v)-5 b(ariance)38 b(of)166 3186 y(\(20\).)23 b(If)g(the)h(conformally)g(in)m(v)-5 b(arian)m(t)24 b(condition)g Fm(A)2051 3201 y Fk(\000)2110 3186 y Fp(\()p Fm(x)p Fp(\))k(=)f(0)d(is)f(imp)s(osed,)i(the)f(constan) m(t)166 3306 y Fm(s)33 b Fp(m)m(ust)g(b)s(e)g(zero)g([15].)166 3738 y Fq(6)112 b(A)35 b(Complete)h(Set)g(of)g(Splitting)f(Relations)i (for)e(the)h(Dirac's)f(Construc-)334 3859 y(tion)166 4204 y Fp(When)28 b(one)g(w)m(an)m(ts)g(to)f(obtain)g(a)g(conformally)h (in)m(v)-5 b(arian)m(t)27 b(DO)g(in)g(Mink)m(o)m(wski)j(space)f(us-)166 4324 y(ing)j(the)g(Dirac's)f(construction)i(and)f(starting)g(from)f(a)g (manifestly)i Fm(O)2788 4339 y Fu(0)2827 4324 y Fp(\(2)p Fm(;)17 b Fp(4\)-in)m(v)-5 b(arian)m(t)166 4444 y(DO)37 b(in)g Fi(R)544 4408 y Fu(6)583 4444 y Fp(,)h(the)f(restriction)i(of)d (this)i(DO)f(to)g(the)g(submanifold)i Fm(M)47 b Fp(\(3\))37 b(pla)m(ys)i(a)e(cru-)166 4565 y(cial)d(role.)g(T)-8 b(o)34 b(obtain)g(a)g(conformally)h(in)m(v)-5 b(arian)m(t)34 b(DO)f(in)i Fm(M)10 b Fp(,)34 b(one)h(needs)g(an)f Fm(O)3165 4580 y Fu(0)3204 4565 y Fp(\(2)p Fm(;)17 b Fp(4\))o(-)166 4685 y(in)m(v)-5 b(arian)m(t)47 b(splitting)g(relation.)f(W)-8 b(e)47 b(shall)g(sho)m(w)g(that)f(a)g(complete)h(set)g(of)f(splitting) 166 4806 y(relations)h(for)f(noncanonical)i(conformal)e(dimensions)j (can)e(b)s(e)g(obtained)g(from)f(the)166 4926 y(prolongation)32 b(of)g(the)h(6-dimensional)h(D'Alem)m(b)s(ertian)f Fc(\003)2372 4941 y Fu(6)2412 4926 y Fp(.)166 5151 y(Let)g(us)g(consider)h(the)f (equation)1569 5380 y Fc(\003)1646 5395 y Fu(6)1686 5380 y Fl(A)1766 5395 y Fh(m)1860 5380 y Fp(=)28 b(0)k Fm(;)1208 b Fp(\(27\))1745 5712 y(13)p eop %%Page: 14 14 14 13 bop 166 83 a Fp(and)33 b(let)g Fi(R)f Fp(act)h(on)f Fm(C)976 47 y Fk(1)1051 83 y Fp(\()p Fm(\034)1142 47 y Fk(\003)1182 83 y Fp(\()p Fi(R)1292 47 y Fu(6)1331 83 y Fp(\)\))g(according)h(to)g(\(8\).)f(In)h Fm(Q)p Fp(-co)s(ordinates,)g(\(27\))f(reads)427 428 y(1)p 404 472 94 4 v 404 564 a Fm(k)458 535 y Fu(2)524 495 y Fc(\003)p Fm(A)674 510 y Fh(\026)744 495 y Fp(+)854 428 y(2)p 852 472 55 4 v 852 564 a Fm(k)932 495 y(@)983 510 y Fk(\000)1043 495 y Fm(@)1094 510 y Fu(+)1153 495 y Fm(A)1226 510 y Fh(\026)1295 495 y Fl(\000)1405 428 y Fp(2)p Fm(\036)p 1405 472 107 4 v 1411 564 a(k)1465 535 y Fu(2)1538 495 y Fm(@)1589 510 y Fu(+)1649 495 y Fm(@)1700 510 y Fu(+)1759 495 y Fm(A)1832 510 y Fh(\026)1901 495 y Fp(+)2031 428 y(2)p 2009 472 94 4 v 2009 564 a Fm(k)2063 535 y Fu(2)2129 495 y Fm(@)2180 510 y Fh(\026)2227 495 y Fm(A)2300 510 y Fu(+)2387 495 y Fp(=)27 b(0)427 666 y(1)p 404 710 V 404 802 a Fm(k)458 773 y Fu(2)524 734 y Fc(\003)p Fm(A)674 749 y Fk(\000)756 734 y Fp(+)867 666 y(2)p 864 710 55 4 v 864 802 a Fm(k)945 734 y(@)996 749 y Fk(\000)1055 734 y Fm(@)1106 749 y Fu(+)1166 734 y Fm(A)1239 749 y Fk(\000)1320 734 y Fl(\000)1430 666 y Fp(2)p Fm(\036)p 1430 710 107 4 v 1436 802 a(k)1490 773 y Fu(2)1563 734 y Fm(@)1614 749 y Fu(+)1673 734 y Fm(@)1724 749 y Fu(+)1784 734 y Fm(A)1857 749 y Fk(\000)1938 734 y Fl(\000)2070 666 y Fp(2)p 2048 710 94 4 v 2048 802 a Fm(k)2102 773 y Fu(3)2168 734 y Fp(\()p Fm(@)h Fl(\001)22 b Fm(A)p Fp(\))834 b(\(28\))589 972 y(+)698 904 y(2)p 675 948 V 675 1040 a Fm(k)729 1011 y Fu(2)795 972 y Fm(@)846 987 y Fu(+)906 972 y Fm(A)979 987 y Fk(\000)1060 972 y Fl(\000)1192 904 y Fp(2)p 1170 948 V 1170 1040 a Fm(k)1224 1011 y Fu(2)1290 972 y Fm(@)1341 987 y Fk(\000)1400 972 y Fm(A)1473 987 y Fu(+)1555 972 y Fp(+)1663 904 y(4)p Fm(\036)p 1663 948 107 4 v 1669 1040 a(k)1723 1011 y Fu(3)1796 972 y Fm(@)1847 987 y Fu(+)1906 972 y Fm(A)1979 987 y Fu(+)2061 972 y Fl(\000)2193 904 y Fp(4)p 2170 948 94 4 v 2170 1040 a Fm(k)2224 1011 y Fu(3)2290 972 y Fm(A)2363 987 y Fu(+)2450 972 y Fp(=)28 b(0)427 1142 y(1)p 404 1187 V 404 1278 a Fm(k)458 1249 y Fu(2)524 1210 y Fc(\003)p Fm(A)674 1225 y Fu(+)756 1210 y Fp(+)867 1142 y(2)p 864 1187 55 4 v 864 1278 a Fm(k)945 1210 y(@)996 1225 y Fk(\000)1055 1210 y Fm(@)1106 1225 y Fu(+)1166 1210 y Fm(A)1239 1225 y Fu(+)1320 1210 y Fl(\000)1430 1142 y Fp(2)p Fm(\036)p 1430 1187 107 4 v 1436 1278 a(k)1490 1249 y Fu(2)1563 1210 y Fm(@)1614 1225 y Fu(+)1673 1210 y Fm(@)1724 1225 y Fu(+)1784 1210 y Fm(A)1857 1225 y Fu(+)1944 1210 y Fp(=)f(0)33 b Fm(:)166 1484 y Fp(After)g Fi(R)p Fp(-reduction)g(and)f(restriction)i(of)e(\(28\))g(to)g Fm(M)10 b Fp(,)34 b(w)m(e)f(obtain)394 1835 y Fc(\003)p Fm(A)544 1850 y Fh(\026)592 1835 y Fp(\()p Fm(x)p Fp(\))22 b(+)g(2\(1)g Fl(\000)g Fm(\025)p Fp(\))p Fm(@)1246 1850 y Fu(+)1306 1835 y Fm(A)1379 1850 y Fh(\026)1425 1835 y Fp(\()p Fm(x)p Fp(\))h(+)f(2)p Fm(@)1777 1850 y Fh(\026)1824 1835 y Fm(A)1897 1850 y Fu(+)1956 1835 y Fp(\()p Fm(x)p Fp(\))28 b(=)f(0)394 1971 y Fc(\003)p Fm(A)544 1986 y Fk(\000)604 1971 y Fp(\()p Fm(x)p Fp(\))22 b(+)g(2)p Fm(@)28 b Fl(\001)22 b Fm(A)p Fp(\()p Fm(x)p Fp(\))h(+)f(2\(1)f Fl(\000)i Fm(\025)p Fp(\))p Fm(@)1761 1986 y Fu(+)1820 1971 y Fm(A)1893 1986 y Fk(\000)1952 1971 y Fp(\()p Fm(x)p Fp(\))g(+)f(2\()p Fm(\025)g Fl(\000)g Fp(3\))p Fm(A)2629 1986 y Fu(+)2688 1971 y Fp(\()p Fm(x)p Fp(\))28 b(=)g(0)280 b(\(29\))394 2103 y Fc(\003)p Fm(A)544 2118 y Fu(+)604 2103 y Fp(\()p Fm(x)p Fp(\))22 b(+)g(2\(1)g Fl(\000)h Fm(\025)p Fp(\))p Fm(@)1259 2118 y Fu(+)1318 2103 y Fm(A)1391 2118 y Fu(+)1450 2103 y Fp(\()p Fm(x)p Fp(\))28 b(=)g(0)k Fm(:)166 2347 y Fp(F)-8 b(or)38 b(the)i(canonical)g(conformal)f (dimension)i Fm(\025)e Fp(=)g(1,)g(the)h(DO)e(in)i(the)f(left-hand)h (side)166 2467 y(of)c(\(29\))h(is)g(in)m(ternal)h(for)e Fm(M)10 b Fp(.)38 b(If)f Fm(\025)e Fl(6)p Fp(=)g(1,)h(\(29\))h(giv)m (es)h(conformally)f(in)m(v)-5 b(arian)m(t)38 b(splitting)166 2588 y(relations)33 b(for)f(\014rst-order)h(DOs)f(on)h Fm(C)1601 2551 y Fk(1)1676 2588 y Fp(\()p Fm(\034)11 b Fp(\()p Fi(R)1877 2551 y Fu(6)1916 2588 y Fp(\)\):)374 2995 y Fm(@)425 3010 y Fu(+)484 2995 y Fm(A)557 3010 y Fh(\026)604 2995 y Fp(\()p Fm(x)p Fp(\))17 b(=)1006 2928 y(1)p 854 2972 353 4 v 854 3063 a(2\()p Fm(\025)22 b Fl(\000)h Fp(1\))1216 2995 y([)p Fc(\003)p Fm(A)1393 3010 y Fh(\026)1441 2995 y Fp(\()p Fm(x)p Fp(\))f(+)g(2)p Fm(@)1792 3010 y Fh(\026)1839 2995 y Fm(A)1912 3010 y Fu(+)1971 2995 y Fp(\()p Fm(x)p Fp(\)])361 3253 y Fm(@)412 3268 y Fu(+)472 3253 y Fm(A)545 3268 y Fk(\000)604 3253 y Fp(\()p Fm(x)p Fp(\))17 b(=)1006 3186 y(1)p 854 3230 V 854 3321 a(2\()p Fm(\025)22 b Fl(\000)h Fp(1\))1216 3253 y([)p Fc(\003)p Fm(A)1393 3268 y Fk(\000)1453 3253 y Fp(\()p Fm(x)p Fp(\))f Fl(\000)h Fm(@)28 b Fl(\001)22 b Fm(A)p Fp(\()p Fm(x)p Fp(\))h(+)f(2\()p Fm(\025)f Fl(\000)i Fp(3\))p Fm(A)2585 3268 y Fu(+)2644 3253 y Fp(\()p Fm(x)p Fp(\)])478 b(\(30\))374 3511 y Fm(@)425 3526 y Fu(+)484 3511 y Fm(A)557 3526 y Fh(\026)604 3511 y Fp(\()p Fm(x)p Fp(\))17 b(=)1006 3444 y(1)p 854 3488 V 854 3579 a(2\()p Fm(\025)22 b Fl(\000)h Fp(1\))1233 3511 y Fc(\003)p Fm(A)1383 3526 y Fu(+)1443 3511 y Fp(\()p Fm(x)p Fp(\))32 b Fm(:)166 3922 y Fp(The)42 b(\014rst)g(prolongation)f(of)g(\(28\))g(giv)m(es)h(a) g(complete)g(set)g(of)f(conformally)h(in)m(v)-5 b(arian)m(t)166 4042 y(splitting)33 b(relations)h(for)e(second-order)h(DOs)g(if)f Fm(\025)c Fl(6)p Fp(=)f(0)p Fm(;)17 b Fp(1:)374 4444 y Fm(@)425 4459 y Fu(+)484 4444 y Fm(@)535 4459 y Fu(+)595 4444 y Fm(A)668 4459 y Fh(\026)714 4444 y Fp(\()p Fm(x)p Fp(\))g(=)1145 4377 y(1)p 965 4421 409 4 v 965 4512 a(4\()p Fm(\025)k Fl(\000)i Fp(1\))p Fm(\025)1400 4348 y Fg(h)1439 4444 y Fc(\003)1516 4403 y Fu(2)1556 4444 y Fm(A)1629 4459 y Fh(\026)1676 4444 y Fp(\()p Fm(x)p Fp(\))g(+)f(2)p Fm(@)2028 4459 y Fh(\026)2074 4444 y Fc(\003)p Fm(A)2224 4459 y Fu(+)2284 4444 y Fp(\()p Fm(x)p Fp(\))2415 4348 y Fg(i)361 4702 y Fm(@)412 4717 y Fu(+)472 4702 y Fm(@)523 4717 y Fu(+)582 4702 y Fm(A)655 4717 y Fk(\000)714 4702 y Fp(\()p Fm(x)p Fp(\))17 b(=)1145 4635 y(1)p 965 4679 V 965 4770 a(4\()p Fm(\025)k Fl(\000)i Fp(1\))p Fm(\025)1400 4606 y Fg(h)1439 4702 y Fc(\003)1516 4661 y Fu(2)1556 4702 y Fm(A)1629 4717 y Fk(\000)1689 4702 y Fp(\()p Fm(x)p Fp(\))f Fl(\000)h Fp(4)p Fc(\003)p Fm(@)28 b Fl(\001)22 b Fm(A)p Fp(\()p Fm(x)p Fp(\))g(+)g(4\()p Fm(\025)g Fl(\000)h Fp(3\))p Fc(\003)p Fm(A)3024 4717 y Fu(+)3083 4702 y Fp(\()p Fm(x)p Fp(\))3214 4606 y Fg(i)361 4960 y Fm(@)412 4975 y Fu(+)472 4960 y Fm(@)523 4975 y Fu(+)582 4960 y Fm(A)655 4975 y Fu(+)714 4960 y Fp(\()p Fm(x)p Fp(\))17 b(=)1145 4893 y(1)p 965 4937 V 965 5028 a(4\()p Fm(\025)k Fl(\000)i Fp(1\))p Fm(\025)1400 4960 y Fc(\003)1477 4919 y Fu(2)1517 4960 y Fm(A)1590 4975 y Fu(+)1682 4960 y Fm(:)166 5259 y Fp(The)32 b(second)g(prolongation)f(of)f(\(28\))h(giv)m (es)h(a)f(complete)h(set)g(of)f(conformally)g(in)m(v)-5 b(arian)m(t)166 5380 y(splitting)33 b(relations)h(for)e(third-order)g (DOs)h(if)f Fm(\025)c Fl(6)p Fp(=)f Fl(\000)p Fp(1)p Fm(;)17 b Fp(0)p Fm(;)g Fp(1:)1745 5712 y(14)p eop %%Page: 15 15 15 14 bop 374 132 a Fm(@)425 147 y Fu(+)484 132 y Fm(@)535 147 y Fu(+)595 132 y Fm(@)646 147 y Fu(+)705 132 y Fm(A)778 147 y Fh(\026)825 132 y Fp(\()p Fm(x)p Fp(\))17 b(=)1406 64 y(1)p 1075 108 711 4 v 1075 200 a(8\()p Fm(\025)22 b Fl(\000)h Fp(1\))p Fm(\025)p Fp(\()p Fm(\025)e Fp(+)h(1\))1813 35 y Fg(h)1852 132 y Fc(\003)1929 90 y Fu(3)1969 132 y Fm(A)2042 147 y Fh(\026)2088 132 y Fp(\()p Fm(x)p Fp(\))h(+)f(4)p Fm(@)2440 147 y Fh(\026)2487 132 y Fc(\003)2564 90 y Fu(2)2604 132 y Fm(A)2677 147 y Fu(+)2736 132 y Fp(\()p Fm(x)p Fp(\))g(+)g(4)p Fm(\025@)3144 147 y Fh(\026)3191 132 y Fc(\003)p Fm(A)3341 147 y Fu(+)3401 132 y Fp(\()p Fm(x)p Fp(\))3532 35 y Fg(i)361 390 y Fm(@)412 405 y Fu(+)472 390 y Fm(@)523 405 y Fu(+)582 390 y Fm(@)633 405 y Fu(+)693 390 y Fm(A)766 405 y Fk(\000)825 390 y Fp(\()p Fm(x)p Fp(\))17 b(=)1406 322 y(1)p 1075 366 V 1075 458 a(8\()p Fm(\025)22 b Fl(\000)h Fp(1\))p Fm(\025)p Fp(\()p Fm(\025)e Fp(+)h(1\))1813 293 y Fg(h)1852 390 y Fc(\003)1929 349 y Fu(3)1969 390 y Fm(A)2042 405 y Fk(\000)2101 390 y Fp(\()p Fm(x)p Fp(\))g Fl(\000)h Fp(6)p Fc(\003)2480 349 y Fu(2)2520 390 y Fm(@)28 b Fl(\001)21 b Fm(A)p Fp(\()p Fm(x)p Fp(\))i(+)f(6\()p Fm(\025)g Fl(\000)g Fp(3\))p Fc(\003)3402 349 y Fu(2)3442 390 y Fm(A)3515 405 y Fu(+)3574 390 y Fp(\()p Fm(x)p Fp(\))3705 293 y Fg(i)374 648 y Fm(@)425 663 y Fu(+)484 648 y Fm(@)535 663 y Fu(+)595 648 y Fm(@)646 663 y Fu(+)705 648 y Fm(A)778 663 y Fh(\026)825 648 y Fp(\()p Fm(x)p Fp(\))17 b(=)1406 580 y(1)p 1075 624 V 1075 716 a(8\()p Fm(\025)22 b Fl(\000)h Fp(1\))p Fm(\025)p Fp(\()p Fm(\025)e Fp(+)h(1\))1813 648 y Fc(\003)1890 607 y Fu(3)1929 648 y Fm(A)2002 663 y Fu(+)2062 648 y Fp(\()p Fm(x)p Fp(\))32 b Fm(:)166 1024 y Fp(In)h(general,)g(the)g(\()p Fm(k)25 b Fl(\000)e Fp(1\)st)33 b(prolongation)f(of)g(\(28\),)g(follo)m(w)m(ed)i(b)m(y)f(a) g(restriction)h(to)e Fm(M)10 b Fp(,)166 1145 y(pro)m(vides)43 b(a)e(complete)h(set)g(of)f(conformally)h(in)m(v)-5 b(arian)m(t)42 b(splitting)f(relations)h(for)f(DOs)166 1265 y(of)36 b(order)h Fm(k)j Fp(if)d Fm(\025)d Fl(6)p Fp(=)h(2)25 b Fl(\000)g Fm(k)s(;)17 b Fp(3)25 b Fl(\000)g Fm(k)s(;)17 b(:)g(:)g(:)f(;)h Fp(1.)37 b(In)g(general,)g(the)g(order)g(of)f(the)i (DOs)e(in)h(the)166 1385 y(pro)s(cess)d(of)e(restriction)i(increases.) 166 1606 y(As)e(an)g(example)h(of)e(application,)i(w)m(e)g(reduce)g (the)f Fm(O)2159 1621 y Fu(0)2198 1606 y Fp(\(2)p Fm(;)17 b Fp(4\))o(-in)m(v)-5 b(arian)m(t)32 b(6-curren)m(t)g(con-)166 1726 y(serv)-5 b(ation)33 b(la)m(w)1550 1847 y Fl(r)1633 1862 y Fu(6)1695 1847 y Fl(\001)21 b(J)46 b Fp(=)27 b(0)33 b Fm(;)166 1997 y Fp(whic)m(h)h(in)f Fm(Q)p Fp(-co)s(ordinates)g(reads) 701 2195 y(1)p 678 2239 94 4 v 678 2330 a Fm(k)732 2302 y Fu(2)798 2262 y Fm(@)28 b Fl(\001)22 b Fm(J)31 b Fp(+)1123 2195 y(1)p 1120 2239 55 4 v 1120 2330 a Fm(k)1201 2262 y(@)1252 2277 y Fu(+)1312 2262 y Fm(J)1366 2277 y Fk(\000)1447 2262 y Fl(\000)1559 2195 y Fp(1)p 1556 2239 V 1556 2330 a Fm(k)1637 2262 y(@)1688 2277 y Fk(\000)1747 2262 y Fm(J)1801 2277 y Fu(+)1883 2262 y Fl(\000)1992 2195 y Fp(2)p Fm(\036)p 1992 2239 107 4 v 1998 2330 a(k)2052 2302 y Fu(2)2125 2262 y Fm(@)2176 2277 y Fu(+)2236 2262 y Fm(J)2290 2277 y Fu(+)2371 2262 y Fp(+)2501 2195 y(2)p 2479 2239 94 4 v 2479 2330 a Fm(k)2533 2302 y Fu(2)2599 2262 y Fm(J)2653 2277 y Fu(+)2740 2262 y Fp(=)c(0)32 b Fm(:)166 2508 y Fp(P)m(erforming)38 b Fi(R)p Fp(-reduction)f(for)f (the)i(canonical)f(for)g Fl(J)55 b Fp(conformal)36 b(dimension)j Fm(\025)c Fp(=)g(3,)166 2629 y(w)m(e)f(ha)m(v)m(e)839 2750 y Fm(@)28 b Fl(\001)22 b Fm(J)9 b Fp(\()p Fm(x;)17 b(\036)p Fp(\))22 b(+)g Fm(@)1435 2765 y Fu(+)1494 2750 y Fm(J)1548 2765 y Fk(\000)1607 2750 y Fp(\()p Fm(x;)17 b(\036)p Fp(\))22 b Fl(\000)h Fp(2)p Fm(\036)17 b(@)2137 2765 y Fu(+)2196 2750 y Fm(J)2250 2765 y Fu(+)2308 2750 y Fp(\()p Fm(x;)g(\036)p Fp(\))28 b(=)f(0)33 b Fm(:)166 2900 y Fp(Applying)e(\(30\))e(to)h(this)g(equation,)h(w)m(e)g(obtain)f (the)g(conformally)h(in)m(v)-5 b(arian)m(t)30 b(condition)1305 3165 y Fm(@)f Fl(\001)21 b Fm(J)9 b Fp(\()p Fm(x)p Fp(\))23 b(+)1759 3098 y(1)p 1759 3142 49 4 v 1759 3233 a(2)1834 3165 y Fc(\003)p Fm(J)1965 3180 y Fk(\000)2025 3165 y Fp(\()p Fm(x)p Fp(\))28 b(=)f(0)944 b(\(31\))166 3524 y([7,11,6,13],)28 b(whic)m(h)i(can)f(b)s(e)g(regarded)g(as)f(a)h (conserv)-5 b(ation)29 b(la)m(w)g(for)f Fm(J)2747 3539 y Fh(\026)2794 3524 y Fp(\()p Fm(x)p Fp(\))14 b(+)3039 3485 y Fu(1)p 3039 3501 36 4 v 3039 3558 a(2)3084 3524 y Fm(@)3135 3539 y Fh(\026)3182 3524 y Fm(J)3236 3539 y Fk(\000)3295 3524 y Fp(\()p Fm(x)p Fp(\).)166 3644 y(Calculating)26 b(the)f(div)m(ergence)j(of)d(the)g(\014rst)h(equation) g(of)e(\(21\))h(and)g(taking)g(in)m(to)h(accoun)m(t)166 3765 y(\(31\),)32 b(w)m(e)i(ha)m(v)m(e)1576 3986 y Fc(\003)1653 3944 y Fu(2)1693 3986 y Fm(A)1766 4001 y Fk(\000)1853 3986 y Fp(=)28 b(0)k Fm(:)1215 b Fp(\(32\))166 4306 y(Com)m(bining)34 b(\(32\))e(with)h(\(21\),)f(w)m(e)i(get)e(the)h(one-parametric)g (family)412 4680 y Fm(@)468 4639 y Fh(\026)515 4680 y Fm(F)578 4695 y Fh(\026\027)664 4680 y Fp(\()p Fm(x)p Fp(\))23 b(+)926 4612 y(1)p 926 4657 49 4 v 926 4748 a(2)1001 4680 y Fm(@)1052 4695 y Fh(\027)1096 4680 y Fc(\003)p Fm(A)1246 4695 y Fk(\000)1305 4680 y Fp(\()p Fm(x)p Fp(\))17 b(=)g Fm(J)1600 4695 y Fh(\027)1643 4680 y Fp(\()p Fm(x)p Fp(\))22 b(+)1904 4612 y(1)p 1904 4657 V 1904 4748 a(2)1980 4680 y Fm(@)2031 4695 y Fh(\027)2074 4680 y Fm(J)2128 4695 y Fk(\000)2187 4680 y Fp(\()p Fm(x)p Fp(\))361 4913 y Fm(\014)g Fc(\003)515 4872 y Fu(2)555 4913 y Fm(A)628 4928 y Fk(\000)687 4913 y Fp(\()p Fm(x)p Fp(\))h Fl(\000)950 4846 y Fp(1)p 950 4890 V 950 4981 a(2)1026 4913 y Fc(\003)p Fm(@)28 b Fl(\001)22 b Fm(A)p Fp(\()p Fm(x)p Fp(\))17 b(=)g Fm(J)1600 4928 y Fu(+)1659 4913 y Fp(\()p Fm(x)p Fp(\))22 b Fl(\000)1922 4846 y Fp(1)p 1922 4890 V 1922 4981 a(4)1997 4913 y Fc(\003)p Fm(J)2128 4928 y Fk(\000)2187 4913 y Fp(\()p Fm(x)p Fp(\))33 b Fm(:)166 5159 y Fp(The)h(more)e(p)s(opular)h(curren)m(t)h(conserv)-5 b(ation)33 b(la)m(w)1528 5380 y Fm(@)28 b Fl(\001)22 b Fm(J)9 b Fp(\()p Fm(x)p Fp(\))28 b(=)g(0)k Fm(;)1745 5712 y Fp(15)p eop %%Page: 16 16 16 15 bop 166 83 a Fp(v)-5 b(alid)39 b(for)e(the)i(conformal)f(electro) s(dynamics)j(equations)f(\(20\))d(with)i Fm(J)2852 98 y Fk(\000)2911 83 y Fp(\()p Fm(x)p Fp(\))f(=)f(0,)h(can)166 203 y(b)s(e)30 b(deriv)m(ed)h(from)f(\(31\))f(b)m(y)h(imp)s(osing)g (the)g(conformally)g(in)m(v)-5 b(arian)m(t)31 b(condition)f Fm(J)3188 218 y Fk(\000)3274 203 y Fp(=)e(0.)166 606 y Fq(Ac)m(kno)m(wledgmen)m(ts)166 947 y Fp(W)-8 b(e)41 b(w)m(ould)h(lik)m(e)g(to)e(express)k(our)c(gratitude)h(to)g(Lub)s (omir)f(Chorbadjiev)i(for)f(the)g(in-)166 1067 y(n)m(umerable)36 b(discussions)h(on)d(symmetry)i(and)e(reduction.)i(W)-8 b(e)34 b(are)g(indebted)i(to)e(Prof.)166 1188 y(Luis)29 b(J.)g(Bo)m(y)m(a)h(who)f(read)g(the)g(article)g(and)g(made)g(man)m(y)h (v)-5 b(aluable)29 b(commen)m(ts.)i(W)-8 b(e)29 b(are)166 1308 y(grateful)j(to)g(Prof.)g(Rafael)g(de)g(la)g(Lla)m(v)m(e)h(for)f (his)h(help)g(and)f(encouragemen)m(t.)i(N.P)-8 b(.)33 b(w)m(as)166 1428 y(partially)g(supp)s(orted)g(b)m(y)h(gran)m(ts)f(DMS) f(9802156)f(and)i(T)-8 b(ARP-003658-071.)166 1817 y Fq(References)166 2150 y Fr([1])71 b(O.)49 b(Aharon)m(y)-8 b(,)50 b(S.S.)e(Gubser,)h(J.)g (Maldacena,)i(H.)e(Ooguri,)g(Y.)g(Oz,)h(Large)f Fb(N)59 b Fr(\014eld)332 2263 y(theories,)32 b(string)e(theory)h(and)e(gra)m (vit)m(y)-8 b(,)33 b Ft(Phys.)g(R)-5 b(ep.)31 b Fs(323)h Fr(\(2000\))g(183.)166 2451 y([2])71 b(H.)30 b(Bateman,)i(The)d (conformal)h(transformations)g(of)g(a)g(space)g(of)g(four)f(dimensions) g(and)332 2564 y(their)37 b(applications)h(to)g(geometrical)h(optics,)f Ft(Pr)-5 b(o)g(c.)39 b(L)-5 b(ond.)40 b(Math.)e(So)-5 b(c.)39 b(\(2\))i Fs(7)c Fr(\(1909\))332 2677 y(70.)166 2864 y([3])71 b(E.)24 b(Cunningham,)e(The)i(principle)f(of)h(relativit) m(y)i(in)d(electro)s(dynamics)i(and)e(an)h(extension)332 2977 y(thereof,)31 b Ft(Pr)-5 b(o)g(c.)34 b(L)-5 b(ond.)34 b(Math.)f(So)-5 b(c.)33 b(\(2\))h Fs(8)d Fr(\(1909\))i(77.)166 3164 y([4])71 b(P)-8 b(.)31 b(Dirac,)h(W)-8 b(a)m(v)m(e)33 b(equations)e(in)f(conformal)h(space,)g Ft(A)n(nn.)h(Math.)f Fs(37)g Fr(\(1936\))i(429.)166 3352 y([5])71 b(F.)40 b(Ba)m(y)m(en,)g(M.)f(Flato,)i(Remarks)e(on)g(conformal)g(space,)g Ft(J.)h(Math.)h(Phys.)e Fs(17)g Fr(\(1976\))332 3465 y(1112.)166 3652 y([6])71 b(F.)34 b(Ba)m(y)m(en,)h(M.)e(Flato,)i(C.)e (F)-8 b(ronsdal,)34 b(A.)f(Haidari,)h(Conformal)f(in)m(v)-5 b(ariance)34 b(and)e(gauge)332 3765 y(\014xing)e(in)g(QED,)h Ft(Phys.)i(R)-5 b(ev.)33 b(D)38 b Fs(32)31 b Fr(\(1985\))i(2673.)166 3953 y([7])71 b(B.)33 b(Binegar,)h(C.)e(F)-8 b(ronsdal,)33 b(W.)g(Heidenreic)m(h,)h(Conformal)e(QED,)h Ft(J.)h(Math.)h(Phys.)d Fs(24)332 4065 y Fr(\(1983\))h(2828.)166 4253 y([8])71 b(C.)25 b(Co)s(dirla,)g(H.)g(Osb)s(orn,)f(Conformal)h(in)m(v)-5 b(ariance)26 b(and)e(electro)s(dynamics:)i(applications)332 4366 y(and)k(general)i(formalism,)e Ft(A)n(nn.)i(Phys.)f Fs(260)g Fr(\(1997\))i(91.)166 4553 y([9])71 b(E.S.)e(F)-8 b(radkin,)69 b(M.Y)-8 b(a.)71 b(P)m(alc)m(hik,)g Ft(Conformal)f (Quantum)e(Field)h(The)-5 b(ory)69 b(in)f Fb(D)s Ft(-)332 4666 y(Dimensions)p Fr(.)32 b(Klu)m(w)m(er,)e(Dordrec)m(h)m(t,)i(1996.) 166 4854 y([10])26 b(P)-8 b(.)62 b(F)-8 b(urlan,)62 b(Are)g Fb(S)5 b(O)1155 4868 y Fu(2)p Fh(N)s(;)p Fu(2)1309 4854 y Fr(-co)m(v)-5 b(arian)m(t)64 b(spinor)c(w)m(a)m(v)m(e)k(equations)e (also)g(\\conformal")332 4967 y(co)m(v)-5 b(arian)m(t?)33 b Ft(Nuovo)g(Cimento)k Fs(71A)30 b Fr(\(1982\))j(43.)166 5154 y([11])26 b(P)-8 b(.)53 b(F)-8 b(urlan,)52 b(V.B.)h(P)m(etk)m(o)m (v)-5 b(a,)55 b(G.M.)d(Sotk)m(o)m(v,)i(I.T.)e(T)-8 b(o)s(doro)m(v,)52 b(Conformal)g(quan)m(tum)332 5267 y(electro)s(dynamics)46 b(and)e(nondecomp)s(osable)g(represen)m(tations,)h Ft(R)n(iv.)g(Nuovo)h (Cimento)332 5380 y Fs(8)31 b Fr(\(1985\))i(1.)1745 5712 y Fp(16)p eop %%Page: 17 17 17 16 bop 166 83 a Fr([12])26 b(V.)21 b(Guillemin,)h(S.)e(Stern)m(b)s (erg,)g Ft(V)-7 b(ariations)25 b(on)f(a)g(Theme)g(by)g(Kepler)p Fr(.)d(AMS,)g(Pro)m(vidence,)332 196 y(RI,)31 b(1990.)166 368 y([13])26 b(A.D.)31 b(Haidari,)f(Conformal)g(quan)m(tum)f(Y)-8 b(ang-Mills,)32 b Ft(J.)f(Math.)i(Phys.)d Fs(27)g Fr(\(1986\))i(2409.) 166 540 y([14])26 b(M.)41 b(Henk)m(el,)h Ft(Conformal)i(Invarianc)-5 b(e)43 b(and)g(Critic)-5 b(al)43 b(Phenomena)p Fr(.)f(Springer,)d (Berlin,)332 653 y(1999.)166 825 y([15])26 b(S.)j(Ic)m(hinose,)g(Gauge) h(\014xing)e(problem)g(in)h(the)f(conformal)i(QED,)e Ft(L)-5 b(ett.)32 b(Math.)f(Phys.)e Fs(11)332 938 y Fr(\(1986\))k(113.) 166 1109 y([16])26 b(C.)70 b(Itzykson,)g(H.)g(Saleur,)g(J.-B.)h(Zub)s (er)c(\(Eds.\),)k Ft(Conformal)g(Invarianc)-5 b(e)70 b(and)332 1222 y(Applic)-5 b(ations)35 b(to)e(Statistic)-5 b(al)35 b(Me)-5 b(chanics)p Fr(.)31 b(W)-8 b(orld)31 b(Scien)m(ti\014c,)g(T)-8 b(eanec)m(k,)32 b(NJ,)f(1988.)166 1394 y([17])26 b(G.)42 b(Mac)m(k,)g(A.)f(Salam,)g(Finite-comp)s(onen)m (t)i(\014eld)d(represen)m(tations)h(of)g(the)g(conformal)332 1507 y(group,)30 b Ft(A)n(nn.)i(Phys.)f Fs(53)g Fr(\(1969\))i(174.)166 1679 y([18])26 b(D.H.)39 b(Ma)m(y)m(er,)h(V)-8 b(ector)39 b(and)e(tensor)h(\014elds)f(on)h(conformal)g(space,)h Ft(J.)f(Math.)i(Phys.)e Fs(16)332 1792 y Fr(\(1975\))33 b(884.)166 1964 y([19])26 b(P)-8 b(.A.)22 b(Nik)m(olo)m(v,)i(N.P)-8 b(.)22 b(P)m(etro)m(v,)h(A)e(lo)s(cal)i(approac)m(h)e(to)h(dimensional) f(reduction:)h(I.)f(General)332 2077 y(formalism,)31 b Ft(J.)h(Ge)-5 b(om.)34 b(Phys.)d Fr(\(submitted,)f(2001\).)166 2249 y([20])c(A.)32 b(Salam,)f(J.)g(Strathdee,)g(Nonlinear)g (realizations,)i(I)s(I:)e(Conformal)f(symmetry)-8 b(,)31 b Ft(Phys.)332 2362 y(R)-5 b(ev.)31 b Fs(184)g Fr(\(1969\),)i(1760.)166 2534 y([21])26 b(I.T.)i(T)-8 b(o)s(doro)m(v,)28 b(M.C.)f(Minc)m(hev,)i (V.B.)f(P)m(etk)m(o)m(v)-5 b(a,)31 b Ft(Conformal)h(Invarianc)-5 b(e)31 b(in)e(Quantum)332 2647 y(Field)34 b(The)-5 b(ory)p Fr(.)32 b(Scuola)f(Normale)g(Sup)s(eriore,)e(Pisa,)i(1978.)166 2818 y([22])26 b(R.P)-8 b(.)93 b(Zaik)m(o)m(v,)i(Conformal)d(in)m(v)-5 b(ariance)94 b(in)e(gauge)h(theories,)h(I)s(I:)d(Quan)m(tum)332 2931 y(electro)s(dynamics,)22 b Ft(T)-7 b(e)i(or)g(et.)25 b(Mat.)f(Fiz.)c Fs(65)h Fr(\(1985\))i(70)f([in)e(Russian;)g(English)g (translation:)332 3044 y Ft(The)-5 b(or.)34 b(and)g(Math.)f(Phys.)d Fs(65)h Fr(\(1985\))i(1016].)166 3436 y Fq(A)112 b(APPENDIX)34 b(A)166 3777 y Fp(\(W)-8 b(e)33 b(use)g(the)g(notations)g(from)f(Sec.)i (2)e(and)h(w)m(ork)g(in)g Fm(Q)p Fp(-co)s(ordinates.\))166 3997 y(The)48 b(manifold)f Fm(Q)876 3961 y Fu(6)962 3997 y Fp(is)g(an)g Fm(O)1299 4012 y Fu(0)1338 3997 y Fp(\(2)p Fm(;)17 b Fp(4\))o(-in)m(v)-5 b(arian)m(t)47 b(submanifold)h(of)e Fi(R)2773 3961 y Fu(6)2812 3997 y Fp(;)h(the)g(action)g(of)166 4117 y Fm(O)241 4132 y Fu(0)280 4117 y Fp(\(2)p Fm(;)17 b Fp(4\))31 b(on)h(it)g(satis\014es)i(condition)f(A)f(from)g(P)m(art)g (I)h(and)f(its)g(tangen)m(t)h(and)f(cotangen)m(t)166 4237 y(lifts)26 b(satisfy)h(condition)f(B)g(from)g(P)m(art)g(I.)g(This) h(yields)g(the)f Fm(O)2411 4252 y Fu(0)2450 4237 y Fp(\(2)p Fm(;)17 b Fp(4\))o(-in)m(tert)m(wining)27 b(SESs)654 4451 y(0)48 b Fl(\000)-30 b(\000)-21 b(\000)-30 b(!)48 b Fm(\034)11 b Fp(\()p Fm(Q)1217 4415 y Fu(6)1257 4451 y Fp(\))1437 4398 y Fh(m)1344 4451 y Fl(\000)-30 b(\000)-21 b(\000)-31 b(!)49 b Fm(\034)11 b Fp(\()p Fi(R)1805 4415 y Fu(6)1845 4451 y Fp(\))1883 4468 y Fh(Q)1939 4449 y Fe(6)2129 4398 y Fh(n)2026 4451 y Fl(\000)-30 b(\000)-22 b(\000)-30 b(!)48 b Fm(\027)6 b Fp(\()p Fm(Q)2492 4415 y Fu(6)2533 4451 y Fp(\))48 b Fl(\000)-30 b(\000)-21 b(\000)-30 b(!)48 b Fp(0)565 4685 y(0)g Fl( )-30 b(\000)-22 b(\000)-30 b(\000)49 b Fm(\034)1013 4649 y Fk(\003)1053 4685 y Fp(\()p Fm(Q)1168 4649 y Fu(6)1208 4685 y Fp(\))1370 4632 y Fh(m)1432 4608 y Fd(\003)1295 4685 y Fl( )-30 b(\000)-23 b(\000)-30 b(\000)49 b Fm(\034)1645 4649 y Fk(\003)1685 4685 y Fp(\()p Fi(R)1795 4649 y Fu(6)1835 4685 y Fp(\))1873 4714 y Fh(Q)1929 4695 y Fe(6)2101 4632 y Fh(n)2144 4608 y Fd(\003)2016 4685 y Fl( )-30 b(\000)-23 b(\000)-30 b(\000)49 b Fm(\027)2367 4649 y Fk(\003)2407 4685 y Fp(\()p Fm(Q)2522 4649 y Fu(6)2562 4685 y Fp(\))g Fl( )-30 b(\000)-23 b(\000)-30 b(\000)50 b Fp(0)32 b Fm(;)166 4831 y Fp(consisting)i(of)e Fm(O)803 4846 y Fu(0)842 4831 y Fp(\(2)p Fm(;)17 b Fp(4\))o(-reducible)34 b(v)m(ector)g(bundles)g(o)m(v)m(er)f Fm(Q)2445 4795 y Fu(6)2485 4831 y Fp(.)166 5051 y(The)d(\014b)s(er)e Fm(\034)11 b Fp(\()p Fm(Q)751 5015 y Fu(6)792 5051 y Fp(\))830 5066 y Fh(b)892 5051 y Fp(of)28 b Fm(\034)11 b Fp(\()p Fm(Q)1167 5015 y Fu(6)1207 5051 y Fp(\))29 b(o)m(v)m(er)g Fm(b)g Fl(2)f Fm(Q)1719 5015 y Fu(6)1758 5051 y Fp(,)h(naturally)g(em)m(b)s (edded)i(in)d Fm(\034)11 b Fp(\()p Fi(R)2959 5015 y Fu(6)2999 5051 y Fp(\))3037 5066 y Fh(b)3071 5051 y Fp(,)29 b(consists)166 5172 y(of)j(the)h(v)m(ectors)h(annihilating)f(on)g Fm(Q)1531 5135 y Fu(6)1603 5172 y Fp(the)g(di\013eren)m(tial)h(form)942 5380 y(d\()p Fm(\030)1077 5395 y Fh(m)1144 5380 y Fm(\030)1192 5339 y Fh(m)1258 5380 y Fp(\))27 b(=)h(d\(2)p Fm(k)1622 5339 y Fu(2)1661 5380 y Fm(\036)p Fp(\))g(=)f(2)p Fm(k)20 b Fp(\(2)p Fm(\036)d Fp(d)p Fm(k)24 b Fp(+)e Fm(k)e Fp(d)p Fm(\036)p Fp(\))32 b Fm(:)1745 5712 y Fp(17)p eop %%Page: 18 18 18 17 bop 166 83 a Fp(Therefore)43 b Fm(\034)11 b Fp(\()p Fm(Q)785 47 y Fu(6)825 83 y Fp(\))41 b(is)h(generated)g(b)m(y)h(the)e (v)m(ector)i(\014elds)g Fm(@)2397 98 y Fh(\026)2485 83 y Fp(and)f Fm(@)2735 98 y Fk(\000)2794 83 y Fp(,)g(and)f Fm(\034)3114 47 y Fk(\003)3154 83 y Fp(\()p Fm(Q)3269 47 y Fu(6)3309 83 y Fp(\))g(is)166 203 y(generated)28 b(b)m(y)h(the)f(di\013eren)m(tial)g(forms)g(d)p Fm(x)1754 167 y Fh(\026)1829 203 y Fp(and)g(d)p Fm(k)i Fp(\()p Fm(\026)e Fp(=)f(0)p Fm(;)17 b Fp(1)p Fm(;)g Fp(2)p Fm(;)g Fp(3\).)26 b(F)-8 b(rom)27 b(the)h(ab)s(o)m(v)m(e)166 324 y(SESs)34 b(w)m(e)f(obtain)g(that)392 544 y Fm(m)p Fp(\()p Fm(\034)11 b Fp(\()p Fm(Q)683 503 y Fu(6)723 544 y Fp(\)\))28 b(=)g Fl(f)p Fp(\()p Fm(X)1108 503 y Fh(\026)1154 544 y Fm(;)17 b(X)1287 503 y Fk(\000)1345 544 y Fm(;)g Fp(0\))p Fl(g)195 b Fp(and)g Fm(n)2131 503 y Fk(\003)2171 544 y Fp(\()p Fm(\027)2263 503 y Fk(\003)2302 544 y Fp(\()p Fm(Q)2417 503 y Fu(6)2457 544 y Fp(\)\))28 b(=)f Fl(f)p Fp(\()p Fm(A)2825 559 y Fh(\026)2872 544 y Fm(;)17 b(A)2989 559 y Fk(\000)3047 544 y Fm(;)g Fp(0\))p Fl(g)166 765 y Fp(are)36 b Fm(O)407 780 y Fu(0)446 765 y Fp(\(2)p Fm(;)17 b Fp(4\))o(-in)m(v)-5 b(arian)m(t)36 b(subbundles)i(of)e Fm(\034)11 b Fp(\()p Fi(R)1894 729 y Fu(6)1934 765 y Fp(\))1971 794 y Fh(Q)2027 775 y Fe(6)2102 765 y Fp(and)35 b Fm(\034)2347 729 y Fk(\003)2387 765 y Fp(\()p Fi(R)2497 729 y Fu(6)2537 765 y Fp(\))2575 794 y Fh(Q)2631 775 y Fe(6)2669 765 y Fp(,)h(resp)s(ectiv)m(ely)-8 b(.)39 b(The)166 894 y(action)22 b(of)f Fm(O)623 909 y Fu(0)662 894 y Fp(\(2)p Fm(;)c Fp(4\))k(on)h Fm(Q)1103 858 y Fu(6)1164 894 y Fp(is)g(transitiv)m(e)h(and)f(w)m(e)h(c)m(ho)s (ose)g(the)f(p)s(oin)m(t)g Fm(y)31 b Fp(:=)c(\(0)p Fm(;)17 b Fp(0)p Fm(;)g Fp(0)p Fm(;)g Fp(0)p Fm(;)g Fp(1)p Fm(;)g Fp(0\))25 b Fl(2)166 1014 y Fm(Q)243 978 y Fu(6)310 1014 y Fp(for)i(a)h(realization)g(of)f Fm(Q)1186 978 y Fu(6)1226 1014 y Fm(=O)1350 1029 y Fu(0)1389 1014 y Fp(\(2)p Fm(;)17 b Fp(4\))o(.)27 b(The)i(action)f(of)f Fm(O)2325 1029 y Fu(0)2364 1014 y Fp(\(2)p Fm(;)17 b Fp(4\))27 b(on)g Fm(y)k Fp(is)d(generated)h(b)m(y)166 1135 y(the)c(v)m(ector)i(\014eld)e Fm(t)850 1099 y Fh(\026)897 1135 y Fm(@)948 1150 y Fh(\026)1002 1135 y Fl(\000)7 b Fm(d@)1188 1150 y Fk(\000)1247 1135 y Fp(.)25 b(Hence,)i(the)e(stationary)g(group)g Fm(O)2569 1150 y Fu(0)2608 1135 y Fp(\(2)p Fm(;)17 b Fp(4\))2825 1164 y Fh(y)2892 1135 y Fp(of)24 b(the)h(p)s(oin)m(t)h Fm(y)166 1255 y Fp(consists)j(of)e(the)g(Loren)m(tz)h(transformations)g Fm(M)10 b Fp(\()p Fm(!)t Fp(\))27 b(:=)h(exp)q(\()p Fm(!)2492 1219 y Fh(\026\027)2577 1255 y Fm(M)2671 1270 y Fh(\026\027)2757 1255 y Fp(\))f(and)g(the)h(sp)s(ecial)166 1376 y(conformal)39 b(transformations)h Fm(K)7 b Fp(\()p Fm(c)p Fp(\),)40 b(and)f(therefore)h(is)g(non)g(compact.)g(The)g(actions)166 1496 y(of)32 b Fm(M)10 b Fp(\()p Fm(!)t Fp(\))33 b(and)f Fm(K)7 b Fp(\()p Fm(c)p Fp(\))33 b(on)f Fm(\034)11 b Fp(\()p Fm(Q)1288 1460 y Fu(6)1328 1496 y Fp(\))1366 1511 y Fh(y)1440 1496 y Fp(and)33 b Fm(\034)1683 1460 y Fk(\003)1723 1496 y Fp(\()p Fm(Q)1838 1460 y Fu(6)1878 1496 y Fp(\))1916 1511 y Fh(y)1989 1496 y Fp(are)395 1890 y Fm(M)10 b Fp(\()p Fm(!)t Fp(\))657 1744 y Fg( )744 1830 y Fh(X)807 1806 y Fa(\026)739 1948 y Fh(X)802 1924 y Fd(\000)871 1744 y Fg(!)964 1890 y Fp(=)1068 1744 y Fg( )1150 1830 y Fh(M)1225 1806 y Fa(\026)1266 1838 y(\027)1305 1830 y Fu(\()p Fh(!)r Fu(\))34 b(0)1260 1948 y(0)144 b(1)1491 1744 y Fg(!)16 b( )1663 1830 y Fh(X)1726 1806 y Fa(\027)1656 1948 y Fh(X)1719 1924 y Fd(\000)1788 1744 y Fg(!)2028 1890 y Fm(K)7 b Fp(\()p Fm(c)p Fp(\))2253 1744 y Fg( )2340 1830 y Fh(X)2403 1806 y Fa(\026)2335 1948 y Fh(X)2398 1924 y Fd(\000)2467 1744 y Fg(!)2561 1890 y Fp(=)2664 1744 y Fg( )2788 1830 y Fh(\016)2821 1797 y Fa(\026)2819 1841 y(\027)2939 1830 y Fu(0)2746 1948 y Fk(\000)p Fu(2)p Fh(c)2867 1956 y Fa(\027)2939 1948 y Fu(1)2991 1744 y Fg(!)16 b( )3162 1830 y Fh(X)3225 1806 y Fa(\027)3155 1948 y Fh(X)3218 1924 y Fd(\000)3287 1744 y Fg(!)3386 1890 y Fm(;)378 2205 y(M)10 b Fp(\()p Fm(!)t Fp(\))640 2059 y Fg( )727 2145 y Fh(A)780 2153 y Fa(\026)722 2263 y Fh(A)775 2272 y Fd(\000)843 2059 y Fg(!)937 2205 y Fp(=)1040 2059 y Fg( )1123 2145 y Fh(M)1198 2122 y Fa(\027)1236 2153 y(\026)1277 2145 y Fu(\()p Fk(\000)p Fh(!)r Fu(\))35 b(0)1260 2263 y(0)172 b(1)1519 2059 y Fg(!)16 b( )1690 2145 y Fh(A)1743 2153 y Fa(\027)1683 2263 y Fh(A)1736 2272 y Fd(\000)1805 2059 y Fg(!)2028 2205 y Fm(K)7 b Fp(\()p Fm(c)p Fp(\))2253 2059 y Fg( )2340 2145 y Fh(A)2393 2153 y Fa(\026)2335 2263 y Fh(A)2388 2272 y Fd(\000)2457 2059 y Fg(!)2550 2205 y Fp(=)2654 2059 y Fg( )2736 2145 y Fh(\016)2769 2112 y Fa(\026)2767 2157 y(\027)2844 2145 y Fu(2)p Fh(c)2910 2153 y Fa(\026)2756 2263 y Fu(0)89 b(1)2968 2059 y Fg(!)17 b( )3140 2145 y Fh(A)3193 2153 y Fa(\027)3133 2263 y Fh(A)3186 2272 y Fd(\000)3255 2059 y Fg(!)3361 2205 y Fm(:)166 2492 y Fp(Therefore)23 b(st)17 b Fm(\034)11 b Fp(\()p Fm(Q)858 2456 y Fu(6)898 2492 y Fp(\))936 2507 y Fh(y)999 2492 y Fp(consists)24 b(of)d(all)g(v)m(ectors)j(of)d(the)h(form)g Fm(X)2459 2456 y Fk(\000)2517 2492 y Fp(\()p Fm(@)2606 2507 y Fk(\000)2666 2492 y Fp(\))2704 2507 y Fh(y)2767 2492 y Fp(whereas)h(st)17 b Fm(\034)3271 2456 y Fk(\003)3311 2492 y Fp(\()p Fm(Q)3426 2456 y Fu(6)3466 2492 y Fp(\))3504 2507 y Fh(y)166 2613 y Fp(consists)42 b(of)d(the)h(zero)g(1-form)f (only)-8 b(.)40 b(Hence,)h(p)s(erforming)f(an)g Fm(O)2646 2628 y Fu(0)2685 2613 y Fp(\(2)p Fm(;)17 b Fp(4\))o(-reduction)40 b(of)166 2733 y Fm(\034)11 b Fp(\()p Fm(Q)334 2697 y Fu(6)374 2733 y Fp(\))33 b(and)f Fm(\034)687 2697 y Fk(\003)727 2733 y Fp(\()p Fm(Q)842 2697 y Fu(6)882 2733 y Fp(\),)g(w)m(e)i(obtain) f(resp)s(ectiv)m(ely)i(the)e(reduced)h(bundles)1240 2954 y Fm(\034)1293 2913 y Fu(v)1336 2954 y Fp(\()p Fm(Q)1451 2913 y Fu(6)1490 2954 y Fp(\))1528 2969 y Fh(O)1582 2978 y Fe(0)1616 2969 y Fu(\(2)p Fh(;)p Fu(4\))1793 2954 y Fp(=)28 b Fm(\034)11 b Fp(\()p Fm(Q)2065 2913 y Fu(6)2105 2954 y Fp(\))2143 2969 y Fh(O)2197 2978 y Fe(0)2231 2969 y Fu(\(2)p Fh(;)p Fu(4\))166 3174 y Fp(and)1095 3295 y(\()p Fm(\034)1186 3254 y Fu(v)1228 3295 y Fp(\()p Fm(Q)1343 3254 y Fu(6)1383 3295 y Fp(\))1421 3254 y Fk(\003)1460 3295 y Fp(\))1498 3311 y Fh(O)1552 3320 y Fe(0)1586 3311 y Fu(\(2)p Fh(;)p Fu(4\))1763 3295 y Fp(=)28 b(\()p Fm(\034)1958 3254 y Fk(\003)1998 3295 y Fp(\()p Fm(Q)2113 3254 y Fu(6)2152 3295 y Fp(\)\))2228 3311 y Fh(O)2282 3320 y Fe(0)2316 3311 y Fu(\(2)p Fh(;)p Fu(4\))2498 3295 y Fm(;)166 3445 y Fp(whic)m(h)34 b(are)f Fj(not)42 b Fp(dual)32 b(b)s(ecause)854 3666 y(dim)1033 3569 y Fg(\020)1082 3666 y Fm(\034)1135 3625 y Fu(v)1178 3666 y Fp(\()p Fm(Q)1293 3625 y Fu(6)1333 3666 y Fp(\))1371 3681 y Fh(O)1425 3690 y Fe(0)1459 3681 y Fu(\(2)p Fh(;)p Fu(4\))1608 3569 y Fg(\021)1658 3592 y Fk(\003)1725 3666 y Fp(=)27 b(dim)2008 3569 y Fg(\020)2057 3666 y Fp(st)17 b Fm(\034)11 b Fp(\()p Fm(Q)2318 3625 y Fu(6)2358 3666 y Fp(\))2396 3681 y Fh(y)2438 3569 y Fg(\021)2487 3592 y Fk(\003)2554 3666 y Fp(=)28 b(1)k Fm(;)166 3886 y Fp(whereas)903 4007 y(dim)1082 3911 y Fg(\020)1132 4007 y Fm(\034)1185 3966 y Fu(v)1228 4007 y Fp(\()p Fm(Q)1343 3966 y Fu(6)1382 4007 y Fp(\))1420 3966 y Fk(\003)1460 3911 y Fg(\021)1509 4061 y Fh(O)1563 4070 y Fe(0)1597 4061 y Fu(\(2)p Fh(;)p Fu(4\))1774 4007 y Fp(=)c(dim)17 b(st)g Fm(\034)2203 3966 y Fk(\003)2243 4007 y Fp(\()p Fm(Q)2358 3966 y Fu(6)2398 4007 y Fp(\))2436 4022 y Fh(y)2505 4007 y Fp(=)27 b(0)33 b Fm(:)166 4437 y Fq(B)112 b(APPENDIX)31 b(B)j({)g(Actions)g(of)h(the)f(Sp)s(ecial)i (Conformal)f(T)-9 b(ransforma-)358 4558 y(tions)37 b(and)i(Dilatations) f(on)g Fm(C)1657 4521 y Fk(1)1731 4558 y Fp(\()p Fm(\034)1822 4521 y Fk(\003)1862 4558 y Fp(\()p Fi(R)1972 4521 y Fu(6)2012 4558 y Fp(\)\))166 4898 y(In)45 b(man)m(y)g(pap)s(ers)g(on)f(the)h (conformal)f(electro)s(dynamics)j(the)d(authors)h(use)g(sligh)m(tly)166 5019 y(di\013eren)m(t)38 b(adapted)f(to)f(the)h(ligh)m(t)g(cone)g(in)g Fi(R)1873 4983 y Fu(6)1949 5019 y Fp(co)s(ordinates,)g(and)g(usually)h (w)m(ork)f(only)166 5139 y(on)e(the)g(ligh)m(t)g(cone)h Fm(Q)1003 5103 y Fu(6)1042 5139 y Fp(,)f(so)g(w)m(e)h(presen)m(t)h(a)d (collection)i(of)f(the)g(basic)h(form)m(ulae)f(for)f(the)166 5259 y(action)43 b(of)f(the)h(conformal)g(group)f(on)h(the)g(1-forms)g (in)g Fi(R)2398 5223 y Fu(6)2480 5259 y Fp(for)f(di\013eren)m(t)i (conformal)166 5380 y(dimensions)35 b Fm(\025)p Fp(.)1745 5712 y(18)p eop %%Page: 19 19 19 18 bop 166 83 a Fp(1.)32 b(Sp)s(ecial)i(conformal)e (transformations:)166 312 y(a\))g(\014nite)i(transformations:)374 664 y Fm(K)7 b Fp(\()p Fm(c)p Fp(\)\()p Fm(A)p Fp(\))731 679 y Fh(\026)777 664 y Fp(\()p Fm(x;)17 b(k)s(;)g(\036)p Fp(\))f(=)h Fm(p)1266 623 y Fk(\000)p Fu(2)1360 664 y Fp([)p Fm(p)p Fp(\()p Fm(\016)1521 623 y Fh(\027)1517 688 y(\026)1586 664 y Fp(+)22 b(2)p Fm(x)1788 679 y Fh(\026)1835 664 y Fm(c)1877 623 y Fh(\027)1920 664 y Fp(\))g Fl(\000)h Fp(2\()p Fm(c)2209 679 y Fh(\026)2277 664 y Fp(+)f Fm(c)2417 623 y Fu(2)2457 664 y Fm(x)2512 679 y Fh(\026)2559 664 y Fp(\)\()p Fm(x)2690 623 y Fh(\027)2756 664 y Fp(+)g Fm(c)2896 623 y Fh(\027)2939 664 y Fp(\()p Fm(x)3032 623 y Fu(2)3094 664 y Fl(\000)g Fp(2)p Fm(\036)p Fp(\)\)])p Fm(A)3476 679 y Fh(\027)3519 664 y Fp(\()3557 623 y Fk(0)3580 664 y Fm(x;)3662 623 y Fk(0)3703 664 y Fm(k)s(;)3784 623 y Fk(0)3824 664 y Fm(\036)p Fp(\))1217 819 y(+2)p Fm(k)s Fp(\()p Fm(c)1476 834 y Fh(\026)1544 819 y Fp(+)g Fm(c)1684 777 y Fu(2)1724 819 y Fm(x)1779 834 y Fh(\026)1826 819 y Fp(\))p Fm(A)1937 834 y Fk(\000)1996 819 y Fp(\()2034 777 y Fk(0)2057 819 y Fm(x;)17 b(pk)s(;)g(p)2352 777 y Fk(\000)p Fu(2)2446 819 y Fm(\036)p Fp(\))22 b Fl(\000)h Fp(4)p Fm(\036p)2820 777 y Fk(\000)p Fu(3)2913 819 y Fp(\()p Fm(c)2993 834 y Fh(\026)3062 819 y Fp(+)f Fm(c)3202 777 y Fu(2)3241 819 y Fm(x)3296 834 y Fh(\026)3343 819 y Fp(\))p Fm(A)3454 834 y Fu(+)3514 819 y Fp(\()3552 777 y Fk(0)3575 819 y Fm(x;)17 b(pk)s(;)g(p)3870 777 y Fk(\000)p Fu(2)3964 819 y Fm(\036)p Fp(\))361 964 y Fm(K)7 b Fp(\()p Fm(c)p Fp(\)\()p Fm(A)p Fp(\))718 979 y Fk(\000)777 964 y Fp(\()p Fm(x;)17 b(k)s(;)g(\036)p Fp(\))f(=)h Fm(pA)1339 979 y Fk(\000)1398 964 y Fp(\()1436 922 y Fk(0)1459 964 y Fm(x;)g(pk)s(;)g(p)1754 922 y Fk(\000)p Fu(2)1848 964 y Fm(\036)p Fp(\))361 1106 y Fm(K)7 b Fp(\()p Fm(c)p Fp(\)\()p Fm(A)p Fp(\))718 1121 y Fu(+)777 1106 y Fp(\()p Fm(x;)17 b(k)s(;)g(\036)p Fp(\))f(=)h(2)p Fm(p)1315 1064 y Fk(\000)p Fu(2)1409 1106 y Fp([)p Fm(c)1478 1064 y Fu(2)1517 1106 y Fp(\()p Fm(x)1610 1064 y Fh(\027)1676 1106 y Fp(+)22 b Fm(c)1816 1064 y Fh(\027)1859 1106 y Fp(\()p Fm(x)1952 1064 y Fu(2)2014 1106 y Fl(\000)h Fp(2)p Fm(\036)p Fp(\)\))e Fl(\000)i Fm(pc)2509 1064 y Fh(\027)2552 1106 y Fp(])p Fm(A)2652 1121 y Fh(\027)2695 1106 y Fp(\()2733 1064 y Fk(0)2757 1106 y Fm(x;)17 b(pk)s(;)g(p)3052 1064 y Fk(\000)p Fu(2)3146 1106 y Fm(\036)p Fp(\))1217 1248 y Fl(\000)p Fp(2)p Fm(k)s(c)1439 1207 y Fu(2)1479 1248 y Fm(A)1552 1263 y Fk(\000)1611 1248 y Fp(\()p Fm(K)7 b Fp(\()p Fl(\000)p Fm(c)p Fp(\)\()1972 1207 y Fk(0)1995 1248 y Fm(x;)17 b(pk)s(;)g(p)2290 1207 y Fk(\000)p Fu(2)2385 1248 y Fm(\036)p Fp(\))k(+)h Fm(p)2649 1207 y Fu(3)2689 1248 y Fp(\()p Fm(p)g Fp(+)g(4)p Fm(\036c)3045 1207 y Fu(2)3084 1248 y Fp(\))p Fm(A)3195 1263 y Fu(+)3254 1248 y Fp(\()3292 1207 y Fk(0)3315 1248 y Fm(x;)17 b(pk)s(;)g(p)3610 1207 y Fk(\000)p Fu(2)3704 1248 y Fm(\036)p Fp(\))32 b Fm(;)166 1486 y Fp(where)42 b Fm(p)f Fp(:=)g Fm(p)p Fp(\()p Fl(\000)p Fm(c;)17 b(x;)g(\036)p Fp(\),)1203 1450 y Fk(0)1226 1486 y Fm(x)1281 1450 y Fh(\026)1370 1486 y Fp(:=)41 b Fm(p)1563 1450 y Fk(\000)p Fu(1)1657 1486 y Fp([)p Fm(x)1739 1450 y Fh(\026)1814 1486 y Fl(\000)28 b Fm(c)1961 1450 y Fh(\026)2008 1486 y Fp(\()p Fm(x)2101 1450 y Fu(2)2168 1486 y Fl(\000)g Fp(2)p Fm(\036)p Fp(\)];)40 b(after)h Fi(R)p Fp(-reduction)g(and)166 1606 y(restriction)34 b(to)e Fm(M)10 b Fp(:)374 1949 y Fm(K)d Fp(\()p Fm(c)p Fp(\)\()p Fm(A)p Fp(\))731 1964 y Fh(\026)777 1949 y Fp(\()p Fm(x)p Fp(\))17 b(=)g Fm(p)1067 1908 y Fk(\000)p Fu(1)p Fk(\000)p Fh(\025)1257 1949 y Fp([)p Fm(p)p Fp(\()p Fm(\016)1418 1908 y Fh(\027)1414 1973 y(\026)1483 1949 y Fp(+)22 b(2)p Fm(x)1685 1964 y Fh(\026)1732 1949 y Fm(c)1774 1908 y Fh(\027)1817 1949 y Fp(\))g Fl(\000)h Fp(2\()p Fm(c)2106 1964 y Fh(\026)2174 1949 y Fp(+)f Fm(c)2314 1908 y Fu(2)2354 1949 y Fm(x)2409 1964 y Fh(\026)2456 1949 y Fp(\)\()p Fm(x)2587 1908 y Fh(\027)2652 1949 y Fp(+)g Fm(x)2805 1908 y Fu(2)2845 1949 y Fm(c)2887 1908 y Fh(\027)2930 1949 y Fp(\)])p Fm(A)3068 1964 y Fh(\027)3112 1949 y Fp(\()p Fm(K)7 b Fp(\()p Fl(\000)p Fm(c)p Fp(\))p Fm(x)p Fp(\))1018 2103 y(+2\()p Fm(c)1223 2118 y Fh(\026)1291 2103 y Fp(+)22 b Fm(c)1431 2062 y Fu(2)1470 2103 y Fm(x)1525 2118 y Fh(\026)1572 2103 y Fp(\))p Fm(A)1683 2118 y Fk(\000)1742 2103 y Fp(\()p Fm(K)7 b Fp(\()p Fl(\000)p Fm(c)p Fp(\))p Fm(x)p Fp(\))361 2252 y Fm(K)g Fp(\()p Fm(c)p Fp(\)\()p Fm(A)p Fp(\))718 2267 y Fk(\000)777 2252 y Fp(\()p Fm(x)p Fp(\))17 b(=)g Fm(p)1067 2211 y Fu(1)p Fk(\000)p Fh(\025)1202 2252 y Fm(A)1275 2267 y Fk(\000)1334 2252 y Fp(\()p Fm(K)7 b Fp(\()p Fl(\000)p Fm(c)p Fp(\))p Fm(x)p Fp(\))361 2397 y Fm(K)g Fp(\()p Fm(c)p Fp(\)\()p Fm(A)p Fp(\))718 2412 y Fu(+)777 2397 y Fp(\()p Fm(x)p Fp(\))17 b(=)g(2)p Fm(p)1116 2356 y Fk(\000)p Fu(1)p Fk(\000)p Fh(\025)1306 2397 y Fp([)p Fm(c)1375 2356 y Fu(2)1414 2397 y Fp(\()p Fm(x)1507 2356 y Fh(\027)1573 2397 y Fp(+)22 b Fm(x)1726 2356 y Fu(2)1766 2397 y Fm(c)1808 2356 y Fh(\027)1851 2397 y Fp(\))g Fl(\000)h Fm(pc)2102 2356 y Fh(\027)2145 2397 y Fp(])p Fm(A)2245 2412 y Fh(\027)2288 2397 y Fp(\()p Fm(K)7 b Fp(\()p Fl(\000)p Fm(c)p Fp(\))p Fm(x)p Fp(\))1018 2542 y Fl(\000)p Fp(2)p Fm(p)1193 2501 y Fk(\000)p Fh(\025)1293 2542 y Fm(c)1335 2501 y Fu(2)1374 2542 y Fm(A)1447 2557 y Fk(\000)1507 2542 y Fp(\()p Fm(K)g Fp(\()p Fl(\000)p Fm(c)p Fp(\))p Fm(x)p Fp(\))23 b(+)f Fm(p)2093 2501 y Fk(\000)p Fu(1)p Fk(\000)p Fh(\025)2283 2542 y Fm(A)2356 2557 y Fu(+)2415 2542 y Fp(\()p Fm(K)7 b Fp(\()p Fl(\000)p Fm(c)p Fp(\))p Fm(x)p Fp(\))33 b Fm(;)166 2780 y Fp(where)h Fm(p)27 b Fp(:=)h Fm(p)p Fp(\()p Fl(\000)p Fm(c;)17 b(x;)g Fp(0\),)33 b Fm(K)7 b Fp(\()p Fl(\000)p Fm(c)p Fp(\))p Fm(x)1491 2744 y Fh(\026)1566 2780 y Fp(:=)28 b Fm(p)1746 2744 y Fk(\000)p Fu(1)1840 2780 y Fp(\()p Fm(x)1933 2744 y Fh(\026)2002 2780 y Fp(+)22 b Fm(x)2155 2744 y Fu(2)2195 2780 y Fm(c)2237 2744 y Fh(\026)2283 2780 y Fp(\);)166 3010 y(b\))33 b(in\014nitesimal)h(transformations:)374 3361 y Fm(\016)t(A)494 3376 y Fh(\026)540 3361 y Fp(\()p Fm(x;)17 b(k)s(;)g(\036)p Fp(\))f(=)h Fl(f)p Fp([\()p Fm(x)1150 3320 y Fu(2)1212 3361 y Fl(\000)22 b Fp(2)p Fm(\036)p Fp(\))p Fm(c)g Fl(\001)g Fm(@)28 b Fl(\000)22 b Fp(2)p Fm(c)g Fl(\001)g Fm(x)p Fp(\()p Fm(x)h Fl(\001)f Fm(@)28 b Fp(+)22 b(1\))g(+)g(2)p Fm(k)s(c)g Fl(\001)g Fm(x@)2839 3376 y Fk(\000)2921 3361 y Fl(\000)g Fp(4)p Fm(\036c)g Fl(\001)g Fm(x@)3347 3376 y Fu(+)3407 3361 y Fp(])p Fm(\016)3481 3320 y Fh(\027)3477 3386 y(\026)980 3507 y Fp(+2\()p Fm(x)1198 3522 y Fh(\026)1244 3507 y Fm(c)1286 3466 y Fh(\027)1352 3507 y Fl(\000)g Fm(c)1493 3522 y Fh(\026)1540 3507 y Fm(x)1595 3466 y Fh(\027)1639 3507 y Fp(\))p Fl(g)p Fm(A)1800 3522 y Fh(\027)1843 3507 y Fp(\()p Fm(x;)17 b(k)s(;)g(\036)p Fp(\))k(+)h(2)p Fm(k)s(c)2438 3522 y Fh(\026)2485 3507 y Fm(A)2558 3522 y Fk(\000)2617 3507 y Fp(\()p Fm(x;)17 b(k)s(;)g(\036)p Fp(\))22 b Fl(\000)g Fp(4)p Fm(\036c)3218 3522 y Fh(\026)3264 3507 y Fm(A)3337 3522 y Fu(+)3397 3507 y Fp(\()p Fm(x;)17 b(k)s(;)g(\036)p Fp(\))361 3652 y Fm(\016)t(A)481 3667 y Fk(\000)540 3652 y Fp(\()p Fm(x;)g(k)s(;)g(\036)p Fp(\))f(=)h([\()p Fm(x)1100 3611 y Fu(2)1162 3652 y Fl(\000)23 b Fp(2)p Fm(\036)p Fp(\))p Fm(c)e Fl(\001)h Fm(@)28 b Fl(\000)23 b Fp(2)p Fm(c)e Fl(\001)h Fm(x)p Fp(\()p Fm(x)h Fl(\001)f Fm(@)28 b Fl(\000)23 b Fm(k)s(@)2366 3667 y Fk(\000)2447 3652 y Fp(+)f(2)p Fm(\036@)2703 3667 y Fu(+)2785 3652 y Fl(\000)g Fp(1\)])p Fm(A)3071 3667 y Fk(\000)3130 3652 y Fp(\()p Fm(x;)17 b(k)s(;)g(\036)p Fp(\))361 3794 y Fm(\016)t(A)481 3809 y Fu(+)540 3794 y Fp(\()p Fm(x;)g(k)s(;)g(\036)p Fp(\))f(=)h Fl(\000)p Fp(2)p Fm(c)1148 3753 y Fh(\027)1191 3794 y Fm(A)1264 3809 y Fh(\027)1308 3794 y Fp(\()p Fm(x;)g(k)s(;)g (\036)p Fp(\))k(+)h([\()p Fm(x)1878 3753 y Fu(2)1940 3794 y Fl(\000)h Fp(2)p Fm(\036)p Fp(\))p Fm(c)f Fl(\001)f Fm(@)28 b Fl(\000)23 b Fp(2)p Fm(c)f Fl(\001)g Fm(@)5 b Fp(\()p Fm(x)23 b Fl(\001)f Fm(@)28 b Fp(+)22 b(2\))980 3927 y(+2)p Fm(k)s(c)g Fl(\001)g Fm(x@)1379 3942 y Fk(\000)1461 3927 y Fl(\000)g Fp(4)p Fm(\036c)g Fl(\001)g Fm(x@)1887 3942 y Fu(+)1947 3927 y Fp(])p Fm(A)2047 3942 y Fu(+)2106 3927 y Fp(\()p Fm(x;)17 b(k)s(;)g(\036)p Fp(\))32 b(;)166 4165 y(after)g Fi(R)p Fp(-reduction)h(and)g(restriction)h(to)e Fm(M)10 b Fp(:)374 4507 y Fm(\016)t(A)494 4522 y Fh(\026)540 4507 y Fp(\()p Fm(x)p Fp(\))17 b(=)g Fl(f)p Fp([)p Fm(x)913 4466 y Fu(2)952 4507 y Fm(c)23 b Fl(\001)e Fm(@)28 b Fl(\000)23 b Fp(2)p Fm(c)f Fl(\001)g Fm(x)p Fp(\()p Fm(x)h Fl(\001)f Fm(@)28 b Fp(+)22 b Fm(\025)p Fp(\)])p Fm(\016)1975 4466 y Fh(\027)1971 4532 y(\026)2040 4507 y Fp(+)g(2\()p Fm(x)2280 4522 y Fh(\026)2327 4507 y Fm(c)2369 4466 y Fh(\027)2434 4507 y Fl(\000)g Fm(c)2575 4522 y Fh(\026)2622 4507 y Fm(x)2677 4466 y Fh(\027)2721 4507 y Fp(\))p Fl(g)p Fm(A)2882 4522 y Fh(\027)2925 4507 y Fp(\()p Fm(x)p Fp(\))g(+)g(2)p Fm(c)3267 4522 y Fh(\026)3314 4507 y Fm(A)3387 4522 y Fk(\000)3446 4507 y Fp(\()p Fm(x)p Fp(\))361 4662 y Fm(\016)t(A)481 4677 y Fk(\000)540 4662 y Fp(\()p Fm(x)p Fp(\))17 b(=)g([)p Fm(x)863 4621 y Fu(2)903 4662 y Fm(c)22 b Fl(\001)g Fm(@)28 b Fl(\000)22 b Fp(2)p Fm(c)g Fl(\001)g Fm(x)p Fp(\()p Fm(x)h Fl(\001)f Fm(@)28 b Fp(+)22 b Fm(\025)g Fl(\000)g Fp(1\)])p Fm(A)2121 4677 y Fk(\000)2180 4662 y Fp(\()p Fm(x;)17 b(k)s(;)g(\036)p Fp(\))361 4804 y Fm(\016)t(A)481 4819 y Fu(+)540 4804 y Fp(\()p Fm(x)p Fp(\))g(=)g Fl(\000)p Fp(2)p Fm(c)949 4763 y Fh(\027)992 4804 y Fm(A)1065 4819 y Fh(\027)1108 4804 y Fp(\()p Fm(x)p Fp(\))23 b(+)f([)p Fm(x)1442 4763 y Fu(2)1482 4804 y Fm(c)g Fl(\001)g Fm(@)28 b Fl(\000)22 b Fp(2)p Fm(c)g Fl(\001)g Fm(@)5 b Fp(\()p Fm(x)24 b Fl(\001)d Fm(@)28 b Fp(+)22 b Fm(\025)g Fp(+)g(1\)])p Fm(A)2700 4819 y Fu(+)2759 4804 y Fp(\()p Fm(x)p Fp(\))33 b Fm(:)166 5151 y Fp(2.)f(Dilatations:)166 5380 y(a\))g(\014nite)i (transformations:)1745 5712 y(19)p eop %%Page: 20 20 20 19 bop 374 87 a Fm(D)s Fp(\()p Fm(d)p Fp(\)\()p Fm(A)p Fp(\))734 102 y Fh(\026)779 87 y Fp(\()p Fm(x;)17 b(k)s(;)g(\036)p Fp(\))g(=)f Fm(e)1264 46 y Fk(\000)p Fh(d)1360 87 y Fm(A)1433 102 y Fh(\026)1479 87 y Fp(\()p Fm(e)1562 46 y Fk(\000)p Fh(d)1658 87 y Fm(x;)h(e)1802 46 y Fh(d)1843 87 y Fm(k)s(;)g(e)1986 46 y Fk(\000)p Fu(2)p Fh(d)2116 87 y Fm(\036)p Fp(\))361 236 y Fm(D)s Fp(\()p Fm(d)p Fp(\)\()p Fm(A)p Fp(\))721 251 y Fk(\000)779 236 y Fp(\()p Fm(x;)g(k)s(;)g(\036)p Fp(\))g(=)f Fm(e)1264 194 y Fh(d)1305 236 y Fm(A)1378 251 y Fk(\000)1437 236 y Fp(\()p Fm(e)1520 194 y Fk(\000)p Fh(d)1615 236 y Fm(x;)h(e)1759 194 y Fh(d)1800 236 y Fm(k)s(;)g(e)1943 194 y Fk(\000)p Fu(2)p Fh(d)2074 236 y Fm(\036)p Fp(\))361 381 y Fm(D)s Fp(\()p Fm(d)p Fp(\)\()p Fm(A)p Fp(\))721 396 y Fu(+)779 381 y Fp(\()p Fm(x;)g(k)s(;)g(\036)p Fp(\))g(=)f Fm(e)1264 340 y Fk(\000)p Fu(2)p Fh(d)1395 381 y Fm(A)1468 396 y Fu(+)1527 381 y Fp(\()p Fm(e)1610 340 y Fk(\000)p Fh(d)1706 381 y Fm(x;)h(e)1850 340 y Fh(d)1890 381 y Fm(k)s(;)g(e)2033 340 y Fk(\000)p Fu(2)p Fh(d)2164 381 y Fm(\036)p Fp(\))32 b(;)166 601 y(after)g Fi(R)p Fp(-reduction)h(and)g(restriction)h(to)e Fm(M)10 b Fp(:)374 916 y Fm(D)s Fp(\()p Fm(d)p Fp(\)\()p Fm(A)p Fp(\))734 931 y Fh(\026)779 916 y Fp(\()p Fm(x)p Fp(\))17 b(=)g Fm(e)1065 875 y Fk(\000)p Fh(\025d)1202 916 y Fm(A)1275 931 y Fh(\026)1321 916 y Fp(\()p Fm(e)1404 875 y Fk(\000)p Fh(d)1500 916 y Fm(x)p Fp(\))361 1068 y Fm(D)s Fp(\()p Fm(d)p Fp(\)\()p Fm(A)p Fp(\))721 1083 y Fk(\000)779 1068 y Fp(\()p Fm(x)p Fp(\))g(=)g Fm(e)1065 1027 y Fk(\000)p Fu(\()p Fh(\025)p Fk(\000)p Fu(1\))p Fh(d)1347 1068 y Fm(A)1420 1083 y Fk(\000)1479 1068 y Fp(\()p Fm(e)1562 1027 y Fk(\000)p Fh(d)1657 1068 y Fm(x)p Fp(\))361 1218 y Fm(D)s Fp(\()p Fm(d)p Fp(\)\()p Fm(A)p Fp(\))721 1233 y Fu(+)779 1218 y Fp(\()p Fm(x)p Fp(\))g(=)g Fm(e)1065 1176 y Fk(\000)p Fu(\()p Fh(\025)p Fu(+1\))p Fh(d)1347 1218 y Fm(A)1420 1233 y Fu(+)1479 1218 y Fp(\()p Fm(e)1562 1176 y Fk(\000)p Fh(d)1657 1218 y Fm(x)p Fp(\))33 b(;)166 1537 y(b\))g(in\014nitesimal)h(transformations:)374 1853 y Fm(\016)t(A)494 1868 y Fh(\026)540 1853 y Fp(\()p Fm(x;)17 b(k)s(;)g(\036)p Fp(\))f(=)h Fl(\000)p Fm(d)g Fp(\()p Fm(x)22 b Fl(\001)g Fm(@)28 b Fl(\000)23 b Fm(k)s(@)1574 1868 y Fk(\000)1655 1853 y Fp(+)f(2)p Fm(\036@)1911 1868 y Fu(+)1993 1853 y Fp(+)g(1\))p Fm(A)2251 1868 y Fh(\026)2297 1853 y Fp(\()p Fm(x;)17 b(k)s(;)g(\036)p Fp(\))361 1988 y Fm(\016)t(A)481 2003 y Fk(\000)540 1988 y Fp(\()p Fm(x;)g(k)s(;)g (\036)p Fp(\))f(=)h Fl(\000)p Fm(d)g Fp(\()p Fm(x)22 b Fl(\001)g Fm(@)28 b Fl(\000)23 b Fm(k)s(@)1574 2003 y Fk(\000)1655 1988 y Fp(+)f(2)p Fm(\036@)1911 2003 y Fu(+)1993 1988 y Fl(\000)g Fp(1\))p Fm(A)2252 2003 y Fk(\000)2311 1988 y Fp(\()p Fm(x;)17 b(k)s(;)g(\036)p Fp(\))361 2121 y Fm(\016)t(A)481 2136 y Fu(+)540 2121 y Fp(\()p Fm(x;)g(k)s(;)g(\036)p Fp(\))f(=)h Fl(\000)p Fm(d)g Fp(\()p Fm(x)22 b Fl(\001)g Fm(@)28 b Fl(\000)23 b Fm(k)s(@)1574 2136 y Fk(\000)1655 2121 y Fp(+)f(2)p Fm(\036@)1911 2136 y Fu(+)1993 2121 y Fp(+)g(2\))p Fm(A)2251 2136 y Fu(+)2309 2121 y Fp(\()p Fm(x;)17 b(k)s(;)g(\036)p Fp(\))33 b(;)166 2341 y(after)f Fi(R)p Fp(-reduction)h(and)g (restriction)h(to)e Fm(M)10 b Fp(:)374 2657 y Fm(\016)t(A)494 2672 y Fh(\026)540 2657 y Fp(\()p Fm(x)p Fp(\))17 b(=)g Fl(\000)p Fm(d)g Fp(\()p Fm(x)22 b Fl(\001)g Fm(@)28 b Fp(+)22 b Fm(\025)p Fp(\))p Fm(A)1436 2672 y Fh(\026)1482 2657 y Fp(\()p Fm(x)p Fp(\))361 2793 y Fm(\016)t(A)481 2808 y Fk(\000)540 2793 y Fp(\()p Fm(x)p Fp(\))17 b(=)g Fl(\000)p Fm(d)g Fp(\()p Fm(x)22 b Fl(\001)g Fm(@)28 b Fp(+)22 b Fm(\025)g Fl(\000)g Fp(1\))p Fm(A)1606 2808 y Fk(\000)1665 2793 y Fp(\()p Fm(x)p Fp(\))361 2925 y Fm(\016)t(A)481 2940 y Fu(+)540 2925 y Fp(\()p Fm(x)p Fp(\))17 b(=)g Fl(\000)p Fm(d)g Fp(\()p Fm(x)22 b Fl(\001)g Fm(@)28 b Fp(+)22 b Fm(\025)g Fp(+)g(1\))p Fm(A)1605 2940 y Fu(+)1664 2925 y Fp(\()p Fm(x)p Fp(\))33 b Fm(:)1745 5712 y Fp(20)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF