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%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: sdp_icml_final2.dvi %%CreationDate: Wed Oct 23 14:02:18 2002 %%Pages: 8 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMBX12 CMBX10 CMCSC10 CMR10 CMTI10 CMSY10 CMMI10 CMMI7 %%+ CMR7 CMEX10 CMBX7 CMSY7 MSBM10 CMR6 CMMI9 CMSY9 CMMI6 CMR9 CMSY6 %%+ CMMI5 CMSL9 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: DVIPS.EXE -P pdf sdp_icml_final2 %DVIPSParameters: dpi=8000, compressed %DVIPSSource: TeX output 2002.10.23:1402 %%BeginProcSet: tex.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S /BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: alt-rule.pro %! % Patch by TVZ % Makes dvips files draw rules with stroke rather than fill. % Makes narrow rules more predictable at low resolutions % after distilling to PDF. % May have unknown consequences for very thick rules. % Tested only with dvips 5.85(k). 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317b3855e3bd8f3621e21d9148bfb669cd05110c3dbb665c668698f460ff82a6 2464b12ede5dc6801f6b35a98645ec804ae4018473df69e7d11422ea4d6f2ff0 f650ac569eae84150cb3e928dde0af9420c00777a47c6ae64a83228855e9aecf 870299662c2503242289a713f440ab4470f0b7adc11ac4b979710a38515d113d 8c58bcd011da70d9f4ee6605470108002d6b91c80c8d34836d734a798dc3d0ed f342aefdd2171c806686adfb0261ef662c82f11a3547dbb4b1e985c30fc5d1a2 8b9f837001d8907e9cc3ba0008e2b2f86b32723eaff107bb363c458fe893c4c3 bfe63f79dcd53641cbd5bc66877d327dc5fc8003fa58dd9088289998bd10de34 b2ac0bc79b592a79b9092917b288902b8d8d4b99838cd8a35ed8c4ef886a3a75 e95d8cad63b3adf91b5b17ceefed215a52288703bf98dbe91ed53134dc30947c 9b245cc426d9dfaf8d2b380eba4f7ac4c8cf006726c8751e9b8beee91284ef0f ce665458e9705125658da64aebc418073c01e297c90574eab86bce31abd9304d 86038b33e71d93aa0fa2025a4ea247609b1c618f505c61a43d03073fbe5dd90a 7ec8ee33bc1b1eebeecebe1b9aa9f0dc1ac641f780dddd4504fe90fe46c3071e 11abf945f1a8a9c4221231e4efebc183495e8314f09d9dba9a9c40389b739e81 8b928fc5256ee5710f9fd22bf983359ab0c0b6a96a479fba1b07402687687ee2 bd992d642768eca67ab641cb342ee95387e045d5d303e019fb12d604a91be5ce a4d9edc06a209a490beef2322571a1fe5772d3102e8acc89506a938251c4d26b 81843e0e2f44db30b0bf107188d556633bcb8c8c15a3fa733e3998290902a77e 87d26bc6382c492f3e447b473702d9afae937ff72e953c9e71a984b563d454af 91f8d1699f2a5b3ab7e1d7c44db3cf86ad3c1ff7399866b32e8534ea829b2809 9f2229f94f3b59c5e56cd1fa9186ea9c186b61c5a45b55ace6298af7a0de581d fc03ad75fc60d25028817f879a6b5c7124df0ce54f5834eee21d04e33ca846ca a94585c90cc201dccaf1d9bd9be86f09a9f025bd99c8533e134899c744b963fe e299a2eaf15732efb2a4731e6e7ec894468f7fb94411ec3c3c1856c2 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI5 %!PS-AdobeFont-1.1: CMMI5 1.100 %%CreationDate: 1996 Aug 02 08:21:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 105 /i put readonly def /FontBBox{37 -250 1349 750}readonly def /UniqueXX 5087380 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311e252234d51741e18db3a68e8ad10242 29e5817a10e796a78d2c7f7c1f50961b9a57aa604c9f821dbf5a9295197bc666 31678d7d2c7e1f8f2151ce0c29efce46270570f4301c5dad1b38884732e53dad 05dfc36b020e726ca575f71f04abf2b49e1f1d6d2a08a477658ca9a48f1c8cce 14382f0a42201de56e2821ee2b0e91a818b1b753d810ec2428dae5153b66af66 9442192024becf743a8f6264a58a0b3aa823510a4957d1bab3e1b83135362888 3e392f49926471a8021fdb48637baf0abcf31204b1b214f48c99f44af7782323 69d772ae4f49be53a1bf584747bf29b94f530d59a5be5370b8dd0691e32a21ba 9d0d592cf3bb50a99a3ecd18d6c4114cb4cfecc49705da90d7edb10e3a28f9eb a49347a09d31eec896f2a28e9607d99d42914aea16545c16e136bdd0b75fe486 61e24c7a9e9b233a76754c760822d0a88c575e08ae3e3f081d7bfb80ec4c03ac a08b65fb6474f71fc95c7c2462f2eb3754e647c29498288f2ced0bd3a2bca5c3 2f80a61fa667802d376cacd34de1f9572d1b3ab09d6f352750db4a768f869638 a87457889aa502850427b40e1294c5ba33e6b17532b2d13b5f50e71582f93daa 231e86597872a2a968127afb0f36257a76ce1eb24e6c4f29fce2aaf6532fb5cf df84da03f03a45b915209f67d398c7c4f7a3b0623750259087a478dca684b29e a8b7f52060bae9394369ed1cecc645d113cca3772a8dc36c6d916b5c5e65454c 0a188a33395fd3da67ad0a31ddab6a8434dd80f0bd0ea1a2a62eecc331d88b3e 369552daa1c575a98fe84383548626b65be7c24ae7eb61159f149872cfd87cf7 6c1a7c155c7186c7600e58cb61b8cb0c5dea98268a39e928bf6fc53985eaa763 2b28c1eb3d82e4d3fe225223f4266d2b6147e344aad2d4884a9308aadc7801af a606838925e3e50ee7aaf73f090baefcfd4e34b795cd92d83c944d91a4940f2f c955d727ea5bd76f0c8057b12621eceb0947b0a5dd10dbcbc52957dd978d7db0 c794660afdd4e946b3d7fc1ba60907fb3f756aab0406c0e7bce5a2238fb3e12e 07e2c0a7f465e90057642d704434ae0b57bd46c8ea91ab2a386e24c6deaf9cb0 8cdaa8fc4ec3fadc06bf28659cb878453db16f4b03925b2127b83be8fc240f33 23ea950b51d70c8fe861e30c346efa0a0125137598caca5478991e35f5bbbc19 4c787488ba339830bca15f8333431902e82b66a514a41c0bff92719a6a0d9587 4cff62c622fc61a0308a69f9fb7ef51bb274ce9e059c04b64e817abdbbb39375 bb4d84ca40de5fa46b84b04f3056d74af428263dccce3d5f6ed4e798b5967d09 f7866e3781a324f3c350b390902a692561a34827acbc45108432ba76005b5036 2b8bd2e9f57f66d4390442ddbb2d60244e11fef8cdfaee58c8b1263bac44e163 e62f1799271e8ea6c2b385c9a2d4fce1184607c7ba642c0f31dfe45b10e673d4 1409cd093292b4b9c34c218a3a3a7881c0902274cf52175380a5352563b93034 09e3d2c5d96a809689326b8c4363d4a34afcf6b8cb0925898fae783ea9c76ff0 70be023fb6b6b096e3cddb1c537efc6bc1db8c99badf033f1d5953e9cee0cce0 607537f0f254d331a0f8559e5347006524cb43509a80f4b81670f1cd7524e34f bebb65689b54a349e58ed6477c3b7eaa5d8666998333ac2104384aa84f44e07d f0a129098c750acc6cf84477f39ab8af3efb5995ab12519172d08af90f7d9db9 48dd64efb6bcdc6d0add1d5f6b697985799de8428e2de45453c44fea 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 161 /minus put dup 164 /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: 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 161 /parenleftbig put dup 164 /bracketrightbig put dup 169 /braceleftbig put dup 170 /bracerightbig put dup 173 /angbracketleftbig put dup 174 /angbracketrightbig put dup 176 /vextenddouble put dup 181 /parenleftbigg put dup 182 /parenrightbigg put dup 195 /parenleftBigg put dup 33 /parenrightBigg put dup 42 /angbracketleftBigg put dup 43 /angbracketrightBigg put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 66 /parenleftex put dup 67 /parenrightex put dup 80 /summationtext put dup 88 /summationdisplay put dup 112 /radicalbig put dup 114 /radicalbigg put dup 115 /radicalBigg put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueXX 5000774 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d5649664203ab24acb938d58d246707ffe7 d62f04bec4b70c21ef75beb2b812622b3c74e969d72d3cd11bd7106294a99caf 0b1629bc7d4de6b96ca82930831d64575f23f4ad06a0e45e315b1d392411be8d 6d73c998789ff258a07a3c8c2057325784514c845500bfd1a971310cfc11d41c 1a167dbd5ff012c60add4e87325f6e5299032a839de65fb1473a166aae1876a4 414a434f22c1d241591fb36f857df6fa930608750ffc0c54f44994662b1f00f1 400bf752ea8d83ffc4cb77a290bc2d99981ae59a191748ba5c7ba1a9d2583fd2 1398452b6ff5d83a059f7eadcd2ef744e9dd22bdf9c79d049bf06835e878c32b 7765c69bdd8ef4deb4ea7cfff4cf9354a4ddffa689de961d16772491c7afbd7f ffde42400764c68e954ee5c455a5687959829bc3b319b2147deaab3628662c80 30c5e02fea09609abe4eaa12e217bc3af673f1bc36a7039eb13fcacb4218fe0f c5a3f9452d4edf46cc91db67b624d4f2d37502fb9f11af4da18ca40b61097f95 d44329375467ed13c5cb585ec53f62b83ef9502cc755af44bf32b87b8ae9f3f2 f8dbf72dab90acafbacd280db6aaffaefdff6d5eff26669bac56280a950560e3 d01714ba38659b0e4e28bf8cdc1f15ff7e3ea9d95a6afbbe3d8e6905aa6d531d 6b6dcab89e2036cc492c65f76ddcc3cfcff33fde3d9ffd10f7a90530fd5c7683 4329baa902e701ca42717476298c9ff511b1941244bb1dfd0e5cab92a43970cd 1bdb292eadd055c97972844fdcdb0aa06401e7b2632b9d134f96bce0064860f3 9859da7a9a82be0a840672a1331212752ae1b521b693b384a72dba8d37a86eaf a0341e7fc5f6ef7a8c566aacb91e8bf5005c18837894c4eacff41dba54e071ca 83fe3c35a5e4206f60b2b4f567b948f7ac54e3ed16b58e137958bb0c141e411b c1c47810add8abfba9f5ef95f96d064cbea94293bf05dc6979c6001553e168fc 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR9 %!PS-AdobeFont-1.1: CMR9 1.0 %%CreationDate: 1991 Aug 20 16:39:59 % 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 (CMR9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR9 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 161 /Gamma put dup 164 /Lambda put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 181 /grave put dup 182 /acute put dup 195 /suppress 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 58 /colon put dup 61 /equal put dup 65 /A put dup 67 /C put dup 69 /E put dup 77 /M put dup 83 /S put dup 84 /T put dup 86 /V 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 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-39 -250 1036 750}readonly def /UniqueXX 5000792 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b448685bdd20eb05d5e7c2126132e33a59a7170609 dcf4871a5d023c9ef57d3362d9f2d7a440bb69bf653364105f16f4d0f03582f9 aced3d05cc76489b16e3fa8a446094d30038b06ecceda269f2eab9d19a99c7f9 39f9548f206c5a457a19270b2b82c43b091dfc5573468eaa3e7a4a32f8042891 d85e4b180fcbcb3091d2800e54c87d84ce9cad6869b5aabbbe47f40c68799893 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0530d7d0495ce477d9d4c964d2440a9913b44081ba9f8bd5d5027a5ddd0bc7b6 8e7344c8e2c3071555b50f3775a6de3041372979b65852919b1e9705f16f326b beaaa02bfec65b720f2fc6ae64ec04417ad9b7ddf169bfc7d50e9845c1cce48c 464543ad8dca6c9ad9453d3a3b9ea76b8aa529b881381abfa4659ef89931284a a68285094440889ee2d37e444353c8fc604d43911beee17a28ee4beeb752af0f 14ca6739d31015a62231dec0c90e6cb546989d8acbe0d0e654797097a769ccf0 91ef99b4c606d0ade980b742307282b7037f55787ed9fbb67893f6fc581d508b d551cf9826dad872ae9110141ee8c13da3ce97b5b601bdb758c19b3912c5cd46 843ba2934aa543ccf250bc82a6a899f50650fc8870584878a594e6a3c72df2de 099868f03cc522e2e87921948552928fdf68b2f50f15749bacac7a771987a440 69c5b8538104761b88b22860b63de255dee2a9b78bd9a289c7fd991e5a64d03d 62e49f0dd89d5c2ae879aa31456b772c69bb171f6206f1d6011d15535f0986fe 2c3886df98bbe883102fda67547426dee061538edaa67b3ba587182eeb812e7a 85b93ed3b958eb57c3459d140cd3ae278a83a1f673c701c20c1cdf0d5c659342 2f5e3f3f9c270dc3ec63ce67fb855fb312c1489b5e93aaf45e1ef306d2d2008d 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 164 /Lambda put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /alpha put dup 175 /beta put dup 176 /gamma put dup 181 /theta put dup 182 /iota put dup 195 /psi put dup 84 /T put dup 100 /d put dup 105 /i put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueXX 5087381 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 dad7b4e6d6f9cf0688b69ba21193bf1495807e7a1e67ed7e41cc25acc04702f6 8ef703e3d45722c1a91fdef7100a48631a02a6f02a08c6b1f9b4df8310385b86 8632718fd87119a233f219d9411383b7fa9f3e4780d8c27e2e89e0cae883d664 c3eac57a3aef8988a2e9f0f8c7f53e0a80bdfc4620e21287d0390e1975398544 7f3ea66401024bea75e1b4c4437b7bb188f76f96b918ac7c6ad7e8ae7f21d8c2 790f08cccec904fe48ef39e597ed4d4237c1d1f596f5906b19ea308020f7a35c 168e327ec3246b1dfabe912f6b6daac09974876d3996e57d180261110db05f15 e3e8eebba3d90b5764c03df3033a1ed678ebc679569a2fb297378b25434c0f20 5313ecb8a952f07242d3ee731b0cdc086a4481178a3d65129c47c09b22e9c431 e11b3747b94c26a757c38d06001798c6a568303d541385244b967d3b1786edea f65bb53c4c2fe75e4b1b15c2c78d930b4296c80f08bad86012451edc8e9f0854 c3b390a16e27b11b3d45a9f72eff8baded2242dc928a61685d79e09681c97425 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25230677c0bec42edfb4089e093aeb4aa0cba6de592c9ecc96a6246b2ecf0b50 3bcc5d339e83da56f142fe7dd904f6c2e142685eac6b48ae4568b932db24176c e788e528458957a2387eeb0e5fbec62fcf97195da4b3651a35295ce98adbda45 ceed833aff11af4acf355b372bb89a4719b56185a63710db50144d9449c892d9 79591947db52df065417c42005b9e939021e6beaa6c86ae2bbe783eab727900e dd9d1b38dfdc45587f320a20fde260e4f9a48405bda805a17ef82c16566195b7 b820b684a259deefb2a97e58bcfef5cc5f5a1e35886782e354a514c5ba457d73 bb5c0f8fd414caf0dcb6eef6c2ffc89fbd5c028f9fd2240fcdd25a768cbfe73f 057ed3f61646787d90bc84c809e9c9494d6043ae13f761689591ac14a983f502 14c7bbe83f2f13ab5ef60cc5fe74b6805565d3014b5110a46e93b1556a689ec2 680b5edbbfde2c4bcd9fa919cd5ef4c3aa79431e4a71c07f360e20dba2cb033b 488670603e3220369730c7166af8659ee8c4dda245882803a1ef82d8c44d40ac 290bcb88ad66da11ae60eb86a84b7dba0cc07b17726aafa262081a007c407cd5 ea8a48a55e4f843e52a803aa4e7b0e1a0c9866028cb7e320f6c80602a60a2b2f 880f6cf0cd598b90a0c92d15ae160c06242d391f7453259a8a6aa9493fc2a6b3 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1c0898fd3508300acfcc63ff1e16ee8c1b60dc9a9cccf822c2d687aff77cce93 27d7001f15975c179b55d4a44f67c1f4de37c5f73d0e66f4a35a349975d95b0a 8ce5ddd217ebe3244e30fe5e1f6f5d69ffe269d84af8485af5fbb5d21df8a1d1 7afa7f928c1820ed8b2446e4af764af6b7195c016fcd868cc355049a29fdbd93 e546b986147475725143f30689ece5aa7b73fa2cc724d149f0d3a3a5d1a46e2c b0801858d38c421ce499cffca0ad8d1c0c6cec0bc4443fad58f816f4063e6db5 d59211a10ea28fce9505f4b8cca3343785a2ea45de854defd89eab546c39b918 b6ac54a64102756782f130ba5b0b05a90f7fff8ae47009902c57f9c6afb8bc1e 8282a3fc551b8456173695536e8e7763cc29c54f010d5ebe8486467206641515 2cceda02b8e8421802155ec877bc9d558e1b9fe5105afbfee19c750942aee977 817831f404dfe9c5ab3c0d177d4bae0524d57732fb1b5746914dc15e89e6d929 f672df6519cac55d795d076f9afb82fe44641f9548cd87ed7ababffd67878459 f0a6eaa45726ab3e78250d98c918e313c65f5918afcdc1d1a463a1d1f1929960 68bf1e2bd2f78d2cad0298f7f03085470535500e9634cb8be3426ebb98cd3194 67b971943b43e38a2f3a351526b8edf5aa103abc9a0a97026ed067543e149d33 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63143fe3ae3d13a9c6c20ff3d8f360e9ded1dfd8af638b625baaed923445206e bc6440b6c46ebcee4e8ea766e13a5c0adbb9d47e2ca19ad5b75761388c935542 0234add83f6f3e64ab66e6feae038f15434a01649ea67f27ff35a25e2565c026 5771a7deed12cf588f076c0b8831316b3772055d490eca9554eeeede9ae58fa6 fc9880a75b8e3545e143e9d2746c8a842888dbb7540c1796466aafc0622b1bf7 9f6c04d643b3c86e0971cbd9b56bbc25d0ec4971d706a15edd34f11ec4ca53a0 8e8fac64f01de5f47feb3f84f4e26dcec9c70ba5462f2ee2bea599680b28746f 9789c90b6af603d2c61c3330b5bd3149116d9e1ad0dbcf3bc383760a36f6c9b4 056bd1da278aabb669387994a772b3a1bd989e2e806f76f61dc3e3eb1c38c7b1 2861169991233a77440ef16fd583f61d5b3dd58f8ec6df21387af13527d4534c d3fab6c213f89a33f0fcfd33f0d8253de0a303b13c67f59e8c8ad4b9ef1a1eb4 0419bb0fdb544d761ce5e011077c82ca35afb80d06b43cbe42be175171d5ce69 e767a6738ffac7cdea8c11b27fbdba81e254196884ac9996a39079339e2abba3 542356334ae73c4e6369d70596e3c3b2d8793f5d4821030b79f21321caa4f48f 9c300a1d7450d4f8d1368b3a390dbf3668a720241f3714cbad770f2627adef8a 2e6aca21be3255d3baed51f190975d19e4e3cc8f42b01691d171d86593aabb0d 35aee2d9e27ca1229478572261ac591bdcebfac06e5bf998c5c4c4fdbd9d69c5 8de1b737822ade25a9007c0c3582292e0c624be9bee0a736f437df1d26137999 58aeb0faa588767153b795c0828321e889f2a9fe461bf1271a202391c3e0e612 f486a098081a5695451598a31615e4e56159d1405d744e49c535cec0a2183e84 7370158f1e89aa9f636737190389c917f37a05165acc6c9e1a2affb824bc4fc6 ba4f194b644578aed95ef3a772bc9d7b30e54ab2a50cd36dda0a199065b2ed0d e932d1163958ab9f86287fbbe1cf735441bd139344f762ee10efd04d9298a279 12dd79107f5985412ad2ad687badf49a1f63c0c646fa86547c790768ad7c5e34 227533787c206faa194e94ee895dd9777d9d7b26996e2145d6d7a60c3e83ca34 7a18fce5f561e02a6ff1fdff0dd045906ab442017b80e60fa09133760274db98 1289762bb5cfbe4ecab6ecc85c19775c7733eda34f1a95d0c57c9572207a4f3e e3e0b14afe300be5c8dedc08e07038958e3b807d9f6f6f3bc6e80b27672b6549 dc5cdcd6bb5b7db36ebc87da8511c990db33f427eca090e446a8f854912599a2 2c24a85ea56f5b7e949ba26c26b555128d724284f56d92b52ced5c8ec82ba64a c5a652b6a6dfd3dd0c8ef7a322022c9fcc7e309f620b895fff45a2f1659a4496 f29cc001c68e54a164ad5e0fd41b4a95ccde810ba83e90cb0984cfa4954d2a6c d24185b3f122fc59719bcffde34231f212b78eb7ae3758e09dd279a92dfc1c32 76ebff930a26ad2de82939 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 161 /lessornotequal put dup 164 /notgreaterequal put dup 169 /lessornotdbleql put dup 170 /greaterornotdbleql put dup 173 /notlessorslnteql put dup 174 /notgreaterorslnteql put dup 175 /lessnotequal put dup 176 /greaternotequal put dup 181 /lessornotsimilar put dup 182 /greaterornotsimilar put dup 195 /notsubsetoreql put dup 82 /R 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 a45b5765f0c1368c0126a1e5d91f20838fde4accf8de587a829ba63d22c9b38d f16a39585bf70bfa77e33c54589b7f610b03133699cfc8e14402612c5ba22d63 8d2f4d4b991d73ff8ebbfab39c7f232c16b80037565a9f26c0be3cd23c617c04 5fd546beb5e96bc6c4c3df11d5b9607c52029e2c33411b164a41866be79520ba 661e0642557aa0a802fcd10c49dd1fe4f83d212d191d1abb4123a3b2f514b9cd 23ded489246938939d52e8248d9821b9aae8f715104e24d15ca01f31b2351165 df9f13eed999d401c98237d77665af89b6ba7483d35014443578e635ac668425 f5e573bc9d30cbabcc7ff2dab3eb2979b54427d968860a3cae1b9abd0083676e 81d496e129af38744263fd3e4953dcb283a16818a1ada003af95b65c30053502 1c55b58ecf4f01db15285c66b9bc0b305bb92b485729c25787e642f2241d92e1 946d30fd995d9d7770d3f5aeab4ca73bd0540dcea43d0f8f9f52bd38228bdf42 7588aafa6078385f292eac65e8dd9214eddb54f13da2e61ae70e7d1426a41e77 938c07a5ce51b62d767898cb7bd9446bc0c08371eea83f25cc42abe9b9e20cb7 2849fa0e497672e4ef67276af626bd69752c1e0724cf700b50d67fdd0a65592f ecec63768b754b40edbbd1beec1445a9125bcbfa1704a7bdb6d11a4770467683 b3159def225201f5183f0b7107108e64bdd309c33361a1ed28671ece26a256aa 9d617a958c3f3b3ba3dd50398b4ab622bdf3ef3123e1ffceed0cc9866e0c4cf9 aa771f48ae44b2be851f350d2c75ac17801ffaa3abf842f040c699cbeaeaf7dd 2901c704ad204ac4329c4c0ab209745c006777d01740049ccc1ff669789befe4 79aa54419cd7f95fb0f41cb15a18dea635854b0a1cfbd94191a8e7af6e463fc8 73b55a3ac5e6eb355cbff7e943f99c6bf461729044efe8c43c73cf7503f26e12 13831e6c7b9b7b8a2f45cfbdfe91d1874d81024f76cfcd2d9821dce625e5e173 09914e0945037997dcaf93712137b8d3080b4ae2f12dfe51b42f1edbff0437a0 eaad35929a9d33f86bd9844ba14cb92fa48cd21937778d7dfeba0dac29ed7933 76b8376067667602bf0b3af948f53138986a3fd59264e656150c2d779f824666 517aa87047ddeb6d74f6f41041b1ac3871cf386094c54cab61b94f71f64796f1 eda3a21d19aa88555b413d67820f02aa68cba02a0174d84547b78f12d17c29fa 94ce89609440b830ceafa2988c6d6529e930f78612cd62b7812198c2c9494924 c5b0f12a3209e161ffa9a7b20326dd1142223d9c409143aa1e39513bf21f5613 bc7cd80426b0dc686df657b3666737c034a35d24f681838c89ded6d867b4f428 22aed116602096b712e4ddf901c7b10615c6b80f6975d01d739a2d7e69680ed4 41ac8d0f9b29464ce2506098c72f42ca6ee4f8a2ab12b8372d27ccf043872e37 3804b08271ab727042d64d3529f31cc9b7bdc838b80f35d6936aaaac30c28353 a8ff773271975cc72821b37533ca1a3a090c980e1f944a3d26cc37ea57db8728 8c7353d2c2a97518b2429d8e34572ae1659b4254b74ed621c1f826d7521d665c ce6b0e9b043aa2fc73e289f5c81375fcffcfdbec05c810b0bb2b06cffe42172e 019336740e7c46e82eb62c1da7fd77331b40f5e11ade360e510f4db160bdef8e 73d850a413c80d38840dcaa3cde3640bac4fc2065bfbc06df2f480f233882b41 1287d3a66fa07891339838aa7528178d0065cbd925e13788ff0d2397587ccd95 2d580752f807a3a9e1edc54b5c18c326675d1c351c2f9422ee7bf6a819b5ab17 0836b8e28134c7d15ecb31f52be8167e21784b0571cb75bd70e7102f051fc8d7 815883c44c19c23dbba2b01d7df4a4b989d98d6c4ed2d5969b265b0a0e002636 b1697fbb612b3abfd021106db2767ac8caa92bd5da09e042de8b08be819f5206 f230b3249a4a49fca617ddc0c521836dc3aa4a2a888a3006e332f80c14618170 e668518908ef6777990ec2b41ac2530ad35158084b46f54d641351677f28c0a6 bd4526fa258f6c0c1aea92a19a34851e7e274a883541bf12cf35b9ed2e48b214 b4bbbc892ee3c636b366418f4c79980a64b491ebb1b22e54d9622b7224bea8f0 645e7eb1b375ab71a66ae323bf9adf599e31119db5ab3c1b9a8e9a9945c936e3 8ecc9b638f95e79159dd527478ccd1d22660c7e82b49ab127619246ef0ce6665 72e236e48c05a9a62b4f332d1785b324d23cb0634c5d294219f2d123ea8795e6 8861b5cb07347ecebbfebeffde98f1aba4a3a961e249061b7d0d55a1abd7a06f 47690109199ca93ae95c2b1932807f0878393e98386ba15ce27d3af00244f091 c473ccf4db00f6b4c65e076bc1c8bffe23010c9e441d05b10ff82d936b0e3077 85057347fa1f969200c803cde2695da3ed57be60e72756c2738bf541ea19ed4e 42ca5572bbd9e8e583b40174da29f5faf9f54cb5ba57f7055e84e76e8d7acb8a f8098981fcb996dc9d562a78191a845758d38855cae07aa55d5db91c6d6abdea d1b5987757b38cc584e4321cca112b7a3181fca4e63572526bdda69d49b51d04 f61a8c03948c9fec61a1dbac3098843ec467f1664d6a0d0aa32dffb7daa20c5d d56dcd6bbed95366dbfeebcee969c85aa70df2a92b76982f3c116e5d850116fd 89a1ae924d8c72d986267b788d2f19a42b94f3ffba2038ed4fa18a0611afef9d f0fe7f38462617becb74895bfee6e31d47ccc72bb97a0dd75765abc959e6a6ad 794fad3ac87961ea09b5ca5967951e570496ec0a4ea5a0252508d11ae109ec65 d5cb11d8b83480df00382cf833d7972f15a8122270b73dd052fc632e0e05b01b e3bcf57af62680698431beee428688ef6a78a2f79e0ff87bf460f59a9d55bc8e f8a431cf8b10bb1d2e5ee01e9bafa7ec3ceaa8e4e5665a3d0c3347fa1d88a555 0c5b9964d10c2be58caac569d183cda6da5345581b3ec3ca95c6ace330325e53 90c4c58105a11ea90fe7fc642f140adbd548639caa5825bca39a2e86d8d82cbc f7638c120bcc883321baefe0db90df1be770c899971eb4258f69b75dcb349adf 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c96410e17e9d2e559c083b4a6c30b5c05261c6e08adb17c884f50f5d05d9e475 39ed2f0a7070a4b0453987cc2088ed272eaed97ed29613c23aff0049c026381b 17c75d94eb548d753e1cf92af19e5b5c35a1157cebec65f6c6312522c9fef524 fb14f44c4fbd8722290f4c9f99544b62336a3f592ea8f5a35b60e17ec04145f5 6cd31cdc2fdac59937956391cb0a7005ebba4f460472edf0ae9fdc9583ccd433 e509c0064abe51bc0ca2657cad4d91df0393f26d4aa1781bbcc952454e2afe02 0ab228b1a22ac318355e7684da0ba6bd05aed01671bc014fcdf154aef62a5058 9dd8faaa1f3268ee152042cb88b444e785821bc076fe71d3ece0aaa6c7f86828 b42e4fc89c519825aa17eba54bb55bf4dcd02a42605ef4282895cebe516459d6 467aa5b1f3cd28a577e16a744aa12653ab1bf1c5ab35eab10e476c504e96e0c8 ec811fa013ea4bba8b0fee9586665ff5e92b1fcc64196f6709879f8df57b98a3 be64d7b9801c012a4d6f7e49f9a2665be9d1bb84d422bdce47802fb788e571df f45b94cbc52ac59cb5f416ce8c3428b6875f8b020ac5ae775a3eb374e7182b53 815cd3b7fbcb2f83a6ac30496af24cb1afc7e0208e4ce849d80e444b3cbc7ba8 614375230633da86179e42df18446ea5fb26b9d982e04e1f00a40c92000d4186 0f28bb531bf3545ed228b1e1ed67daa23704657a9db710e7bf6cdc6994cf96a1 80f6565e8a1452524d8b50aad92daa93b2e39fd8fea8900334350c7ff110b6d3 5da71ec8afb1a08f387420ca1509d80a3a95057a2e0ec55f5fb7f3d8a7df437c f68bf77832e6a683a3e54bc5437b74b2abfa87903ca0c4ec58d45e282c601d22 9bd2f2785c6737d70a84954580d32aeb0ab7d25d83894090d18699001a743f7c 1d9ca616a3d5015010fa22e3f73307bb546f4bcfc526b0540fecffc3a1121ca5 d21e7039d8aa502979f6addae985de0a5bf9518faeae1f0da6d72e763a38aa2a 8bb427c5c48959e6630118a24060e8cf9b12657c650f859ebb4252c8b8f699a0 2a716035e1ed7653bd737500 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY9 %!PS-AdobeFont-1.1: CMSY9 1.0 %%CreationDate: 1991 Aug 15 07:22:27 % 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 (CMSY9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY9 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 161 /minus put dup 164 /asteriskmath put dup 169 /circleplus put dup 170 /circleminus put dup 173 /circlemultiply put dup 174 /circledivide put dup 175 /circledot put dup 176 /circlecopyrt put dup 181 /reflexsubset put dup 182 /reflexsuperset put dup 184 /greaterequal put dup 186 /followsequal put dup 194 /follows put dup 195 /arrowleft put dup 41 /arrowdblright put dup 50 /element put dup 54 /negationslash put readonly def /FontBBox{-30 -958 1146 777}readonly def /UniqueXX 5000819 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f67052788e6703bb5ee49533870bca1f113ad8 3750d597b842d8d96c423ba1273ddc63eb43f34fa90ea73a234af35fd9b24eb7 4f19626e9124df7acb92629cc87b2cf81c4a89b27447806db34393ddf0402958 e2daee0b4bf315f9aa0a2650d2e11ee289ffaec940d0af160fe7f29411c1c458 69d1ac3bcfe2ae4da72422549095f1fc9fec606289316f106e719a4ae2fd45ee 868fddf07a80e9a626bb43cdfdaa9dcd208b88fd99af20b27f9d6b5cda6a3fa8 8d5acfce8b5eebb38854378cecc3bfdac526514f1be9fb28ae628918ba942a3e 05b78c3aff808cf416a5ed01be280231b75a7aee0c66b0549713bef6b3cbc188 e625ba82766494cd8dc214a666d7d12cb23ec030318b42d6c4717b3569194d58 e17059e46f000a600b2be3c532a1ed72f2556490fe85b4fd8a4f49c424df0b54 bd10fcd876a38c3c474e0061fa6b8effefc1503ce1c8e8bc31eb7a0517eef89d 2b77a072b4c67e6c17ceba07a91173bc121affdb29ff7bc1c81a5efe5f3306f6 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI9 %!PS-AdobeFont-1.1: CMMI9 1.100 %%CreationDate: 1996 Jul 23 07:53:55 % 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 (CMMI9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI9 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 161 /Gamma put dup 164 /Lambda put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /alpha put dup 175 /beta put dup 176 /gamma put dup 181 /theta put dup 182 /iota put dup 184 /lambda put dup 185 /mu put dup 186 /nu put dup 194 /chi put dup 195 /psi put dup 58 /period put dup 59 /comma put dup 65 /A put dup 66 /B put dup 67 /C put dup 75 /K put dup 83 /S put dup 88 /X put dup 99 /c put dup 120 /x put dup 121 /y put readonly def /FontBBox{-29 -250 1075 750}readonly def /UniqueXX 5087384 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 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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 161 /Gamma put dup 164 /Lambda put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 181 /grave put dup 182 /acute put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 194 /Oslash put dup 195 /suppress put dup 49 /one put dup 50 /two put dup 51 /three put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueXX 5000789 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 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355cd883742ec9f1846f70000e05282afe8caa264f1042ff2a495f3a0139786b e9fb3a895b0c0d321220ceaeb53bfe22f11700d70a34a171c5d87efaee0c2643 49966f13fc74972fd980522cddd68790efdeea013f696f63410a9cb293ae333f 784f60993e94b5fd175afd4c290efcd2a9c4b042ed17118728b19acc2043a37b 4ac532810cefa01a3da95f0941fca0f33906220915163942610f2b1321add1b4 beda2dfb416c0275c4a0758c2b928d4ae0c0e12a5793baaa5cf221324aa586b8 4873993175d8b9b1afe9f8a3d8f3696bdc10ced4cc3caffaff04ba74978a178d 4f8dbc6ff55b8a290e2149e4f608e35b282192849b3e3fe1b6cd1e37d28457fb 475f9d941e5248b33b22992dbc598120bbfcefe300596cf0778ba20a4ddfb1cf ac3812ac8c0a4315ec4efef197b7da8ae3b3a4ce1eb1f8ef69365cb0e58c5559 1169f4a09b0b48997b0fc52a5676f5eb3eb137b00c65867d76078901f449baac 3005b6bd71bfce1eec4ce8891f66df0df765fa2cc1d910f14cec7adea9fe4613 2f1cd621268e778cfa9db87c2817ea4e629831c782046a7718703bc6a787038e aafb1339dabe08b9452ea31ca4f5014a4a2cbd949faf9d2f135e12af920ba9ac 95b77ede9484462e6b53df87442a588b3e1ea8f7618d47ffd6cef66db6963033 34228de76c0eba8821c6a7f1b3c6aa5f2b8f4626bd93cf4d475d2ff1e74ef84a 2d2d90697517268116ffc831f70aca8a536c7fe85ebc627269114f5b075a6491 d0fde7c1ac3e04cfcb0de221fe27444bd666a097d211e8a0323fa651cbb7f7a9 5516fe6d5ee20ee703f1ad1c205e45d9845ad224d946cc7158222be836a0f166 a67c42c618f07a7f7d244b86bc270c7aec8a14f1dd28bc49f62125501c55c609 cb82dcb808acc91ffea0c13ee95d3ccdab28bf998a5c18bf1db47077ee88974a 67 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /minus put dup 163 /multiply put dup 164 /asteriskmath put dup 169 /circleplus put dup 170 /circleminus put dup 173 /circlemultiply put dup 174 /circledivide put dup 175 /circledot put dup 176 /circlecopyrt put dup 181 /reflexsubset put dup 182 /reflexsuperset put dup 183 /lessequal put dup 184 /greaterequal put dup 185 /precedesequal put dup 186 /followsequal put dup 194 /follows put dup 195 /arrowleft put dup 48 /prime put dup 50 /element put dup 107 /bardbl put readonly def /FontBBox{-15 -951 1252 782}readonly def /UniqueXX 5000817 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f73e9b697ddc0351763500e23c4d170ea8a2d2 75ad0b6b23097f02fa803c1f46f9ac10fe0e527c431b11b1b9a50e874896cca6 17fe3e255960846bc319ca68b9b930a45d5d0fc36b352c3443ab6f7793b38d2c 7fb9c409ab7980ead55e9f8f6b83c70088d1fa6d7f2214da8928e40b96334ad2 72d1b5db9d1355a579b26a8dac21f021e393d7d837a95deb8de92ed818a8b80b 98ee9cdd8a3c57ad7d6eb7a82f43eaa6ec520eefa7302e750edc4b495f70bba2 24a2182eec37f1a1c5dc8e20973007150237daa6bf03d498826777eeebe09014 ebfe05a8731e1a421cfc513d47d673e3d94d805f5d0668ffa265e0962cb5a75a 4a93d9b872d3887398a1240c8c97683ecb8177b5345ef766344188ddfee88fa0 529971cb893d74b4f495d29f9b2c79073c69ef353833b7708112984aa0102f42 9d63603a7a17e83303ae90b4b7ad6851757321568980b30eff7196a3f9cf1d4b 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX7 %!PS-AdobeFont-1.1: CMBX7 1.0 %%CreationDate: 1991 Aug 20 16:35:49 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX7 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 161 /Gamma put dup 163 /Theta put dup 164 /Lambda put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 194 /Oslash put dup 195 /suppress put dup 119 /w put readonly def /FontBBox{-55 -250 1289 751}readonly def /UniqueXX 5000765 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be79011d1f5bfae5c1f476ee6f05eb1d2caeb 269958b194521197b1ee2d9de5f5346241e154574f14a4532b35968596ef7a20 102dad629475cf5d654585d4caac68ba7b2ff21b103a98af3e9ec01fae180804 26b73d62e93f0328ce7775e81898de587dd0fbcfe98934a58afb1aa4dd8c6eec d19373ddd360a5e85cdd4b752b2f2baa13b48a282214a05632d28e915ec90824 21ff8f2166e940c7122c583f235bfd098f8aac0f1dbb9a625b02bd889869664e f1856608899ba509d6eb8ece665afc34a459dd5b54dc7711f6f04a288214efe4 53095e239bf19cd36b7028d46c07a2b8c17d452d8563ea88a3bc5e5ea29a9ea9 a7e38481679f893f9e6a983ccb5512bed4aca5888793526701796b3fcac0fb62 f69c66d0688eede0fefd590fbce954b2cdd2abc9017390077a2cf6f6fa7d0db2 dcb4b0963e3dfbf02be1dca9cad104f3d29495d19337fbffdd745e1729e7c0a6 41fdd3bf42c30d715670b3ade44f481d795071f1b24f4f03dff59d91bf9d678b 4560da27ad826c9fc8f8cd9a333248cb2007b4ac0a9e2d5c044774124f1ee23c 0e3e005a4a16e1d0cba15797fc42fa75cb45b24fcf8a8cf41de8e06388b5f995 04454646c3246074f495564cfc8ad75a949cc477ecf03f0511b90b6a7456b730 cf09773b5544f6ec17a9a179714e47e2eb1142d8005bd228affaa375eed7fb1b 075b3023be6386f3805aee05b47c7f370e8ad410110429d2ee8801d0d2da9e68 a453a323c147716ca588fbf563175e5f47d1086a21b04102cba338a3580225ce 00dfb1784fbcad0fcd11434afcc2ec13337ee24775c2952be02e20572aaab17f d21287775e4a4d9fbf877a487b252b5c880463541489f5578c7472d39fc27320 2e4ec86f9935ab2ded14ab9ea36770d79a1a72161102412bc57e2c2ea4d3a0ca a4d0f64ba95b18d07f2641ae6f920d4495a4ecbb0ce90fe9a4b5e37acc475f60 f67c319d408a832bacf013e9174ae51ae81fe6f3b2b5fb63b327ef6e0b8b9213 da9952ff2ee9fefcabf3ae1685e774338a45d3f066948555a1ebf63f298a3ba5 579025e0aaa6ae449280684e7126d3258712a9d72ace7f604d44024ba4c88da3 1b6c94bc370bde790adcbb55bd895c478237fc6b2886f6af40272ead13186169 bac9a65fbc88702627bfcb27618d7591436a6db90227fb5098211e5cd045d96e be16ac1aec2b5ea1846603a017970d51a4d8cbeadfc101654ca9d27557eb69b1 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33791eac78344a1129c089e030a3816ea9f1280cdcff1ad5b4aa5fc65a09cddb 0934590fb3631edbeceef40585e93719243f3c12fe229c3d43967d49a4747671 094cf7d2ddbbb58ab533082f1ad53f86d52deba632b0f1ea7e019cfc5267ed91 0ae56eab584d3c80a3c18eabf5139aa2c3993521926e569c0ac04cd30a9743cc 752504639afb7a045568f78629e668a968f8231bbc1ea53346b51d2380a8f966 130bb9d4e1efc900f356bc9a0df85cd3e5d2e6a08ab4b0acc10f814dd154c06d b86bf0b91a6eb38cc23b569e6b11e774404a6051bde8e4d3124c1ecbaa792f7e 1f923a5429092eeae11c689114b9a94a0e4628b61875cc0b6ea28519f1819179 9718d616be78061b6e26fee3c7fde470c1416d3ef89b0637795b29226dc9d83d 7a4395e2e70e00e8abb19741e6cfa5de88be78a482a33e7263ebaad31c106737 ac3dc45f2b695f638b1b8d09ddc579de4da81295759a055e52bcea4ff7d3d92d 454d81f96196111ad802b77032716320559c22580088249c2ec9ddf0aa765197 ecb7b01657661666683ec4b1a4f29b4e355cebada142dd2e383e30620152a2ee fda7071d838d40ca4576931006745a418afc9c85b8802b4b203e8308fa73d02d ad4a0625f2f040f7019a2058ce0c037f0cfb92cadc6f465146f73f4cbdeaf123 07a5a006b151db6202d61a12f5ba899002ccda20cf660bc71b3782a4ac23ae21 c31476d6abe7e2e1e07b3ee37bf2d3919b3323625218d3f2ec34a7ec03e80c7e 20c01b7454f583e3df2ec04bfecfb9eb8e32a01363e0fe1355845734b01d34b9 137cb430a2803b07bbd275a9d392d5c6150723b97762d0336e8d230cf65a0e8a 99095cf1888e2aa26871d499ab333e74fec0989e8e380e99e1c817fb37c72807 b71768b6d8e115a6e279a4a4f3b562a1e443808ba2bf12b650d7a73a645ab34a 1b8643d649039b2d1c060a0d29c2e847a589a82e3dbfff5e3a06424e14855b5c 89c18550aee76779c3b84b72bcf590f50b395638f404ed258d0acf99cad3a7d3 9472775e149063b0a0fba756853e1410348bba598ecba9da2717da7256d53e1a 961fbbee72ec5db82399e411890ccf2848761de3dde8688696940fd7acfe6e7c aa9c81a97fcd1ee253ea93dd34c31155de48f11415bd2a4c3a855164314d84d2 104274defae566e253078dfa4c12ea5297628d82fd4dc518f22a61114ea2fbb7 42fa208df533bd0d93d4416634827cf4038224cfa8655f186047405f38cb449c 4e500685a230f7ec3df2e200a3a3051dbbfbe021a0164112e931eb0fad164e67 74a1556d648370dd8d182494b4b88f3d8c8deee0f130f95471c84c9a3b44c9fc e5859e670cef834f4006c0da82cb8b2ae74374e92bcb9164415803b3b3d117ec c7977d174a82e9fe7ed7ff141787fdaa2c621b0120643fec6ea82fbc7be3aaa2 5b25fc94366ff082d162d83d42d74c300a4553b22824b3cad5868db00bda8d7b 86459a3c3ccd7d 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 163 /Theta put dup 164 /Lambda put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 194 /Oslash put dup 195 /suppress put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 61 /equal put readonly def /FontBBox{-27 -250 1122 750}readonly def /UniqueXX 5000790 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b448685940729e6e22f4be0e08c6505cba868f7950 93f556b64b808dd710eb936d3ac83e5783b5f7e56d363a59a026a5619a357d21 c4e77bea121eb24b09027d1aad93158edf912940e91cd2f17922b35885894644 7773730bb19af9e70453e1b78d5edc123b273a979891d6d5e624f4a388f3897e 13b045e34b4ee2f5e4fc5e996d3549640010c671de14d18b0b3cd300e57dac03 22e59bfbf7b29422230870f5897fcfaad4b50c7c1c58edcd119218163d6f6e56 25ccb882db7b6d09a1586508e8cb642a11c29a46915e4a96e282079cb21922c1 c2e360b487a45302fd22ec8c5fab63e54b5e844d4b17ca2fff37c69c366dd72a d02922c14c0932f65ed03e4219c117962edbad2dcdeaa9c10ce8af38a4ae52e2 b377245b0be19a77d6c936e7530cc4d0b78d0cc4a92698fa2870fa54f2d8503e 2d17b3d52fb2febb09f2b2af0c2a1892039ebe19a690098799a858e3d39631bd 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56f6fb9a119b7260bdb75932c8f6c139ae4184cfe08f6046189bbf2900dad6a9 8b16dbce44c657411a512c2ea5fe13cd69f5bab0afca33a285577bd220c9b1e3 636ae76e22641e258763b833327903180e6f9b0f82165f6b8cc228d7e7626f64 6ae22127591ad70d2655524502307c703f44a2d4c8b7880bd21785edcbe44a20 b3229c83876c3d5b278ec2e28aee0074 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 163 /Theta put dup 164 /Lambda put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /alpha put dup 175 /beta put dup 176 /gamma put dup 181 /theta put dup 182 /iota put dup 183 /kappa put dup 184 /lambda put dup 185 /mu put dup 186 /nu put dup 188 /pi put dup 194 /chi put dup 195 /psi put dup 59 /comma put dup 65 /A put dup 66 /B put dup 70 /F put dup 71 /G put dup 75 /K put dup 78 /N put dup 84 /T put dup 98 /b put dup 100 /d put dup 102 /f put dup 103 /g put dup 105 /i put dup 106 /j put dup 108 /l put dup 109 /m put dup 110 /n put dup 112 /p put dup 114 /r put dup 116 /t put dup 119 /w put dup 120 /x put readonly def /FontBBox{0 -250 1171 750}readonly def /UniqueXX 5087382 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930f01bdaae649990604a33aa9eaffbe5b5489 e5c7c9ff9d9be01b08220832c41caac64816b53bbc087ae4b621d2b44b21ae5e 3f7fe4ddb05c675abfe30510eee4a7c874bb57b2ffe521a6875eddfdfd18c781 25bfca5a097aad361dd94df46f14026c25ea988194899b95001209cb7009cead 4b4ea03473ea120e7a7e341d4e3926796347fdb3a124c29660ac09f4648b313c b58e824884382f24ce08d2edc686923acace1201c16a07a42803288cd0b946f7 546510e95b1fa1cc96f0bf72aa823d8e83d7c68c752c82a54c752eed2b1e699d e9db1830272ffbf2f4996ccc2d6fe2ae272798989525ef3b67b0d09bffcef749 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d71a5edb01634f0670ef9e4876ff63511d6390557eab5f57df96ee6144207c68 4a541d51c9e14a22cca83be5376a56d3631f07a49b9dcea4438109d854ef2964 146ff8faede3 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 163 /Theta put dup 164 /Lambda put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /alpha put dup 175 /beta put dup 176 /gamma put dup 177 /delta put dup 178 /epsilon1 put dup 181 /theta put dup 182 /iota put dup 183 /kappa put dup 184 /lambda put dup 185 /mu put dup 186 /nu put dup 188 /pi put dup 190 /sigma put dup 193 /phi put dup 194 /chi put dup 195 /psi put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 65 /A put dup 66 /B put dup 67 /C put dup 70 /F put dup 71 /G put dup 73 /I put dup 75 /K put dup 79 /O put dup 82 /R put dup 83 /S put dup 88 /X put dup 89 /Y 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 105 /i put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 116 /t put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueXX 5087385 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930da4f8d58abefd45db119b10eb409dd89792 3c6e705479464a4b310b58348c4b42393988fef4925cf984423aaf65fea9f0e6 4629953bcf50b919d968d99bd185f83112b2759cc411764e9bde677f57c5ee5a c555448d5f81a16259ded1e11bf4119d53e8ab07a802df900d3a2d5ccc1c6876 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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 161 /minus put dup 163 /multiply put dup 164 /asteriskmath put dup 167 /plusminus put dup 169 /circleplus put dup 170 /circleminus put dup 173 /circlemultiply put dup 174 /circledivide put dup 175 /circledot put dup 176 /circlecopyrt put dup 177 /openbullet put dup 178 /bullet put dup 181 /reflexsubset put dup 182 /reflexsuperset put dup 183 /lessequal put dup 184 /greaterequal put dup 185 /precedesequal put dup 186 /followsequal put dup 188 /approxequal put dup 190 /propersuperset put dup 193 /precedes put dup 194 /follows put dup 195 /arrowleft put dup 33 /arrowright put dup 44 /arrowdblboth put dup 50 /element put dup 60 /Rfractur put dup 70 /F put dup 75 /K put dup 76 /L put dup 80 /P put dup 88 /X put dup 89 /Y put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 112 /radical 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 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7ac272d87243c1fd3950744454dbfe49e39a1b4f7d7240b52675909dc5ef1889 1060b680e51b3239d0cd30aac22e2ec031b6162a1b14bf555f407e17812c16a6 dffabc320a2b6f30b21c574d25fd70135e6cf14706c0087f5cfb2461cc06fcfa ff177afe8f590ab7a1194ad1c606e3301f57216a10c473e76606ca4c94bae677 bf9610973e367d6247d75d792deeec5749f818b7007ce0d2cfe2655ae8f8ad2f 8d931473dfbef6b349b0500b57df545e3715c9a961ba44b04caf01a5d8f7e36b 446c7c61b00cddee9652cf5e9e462451fba046611be3595548fb8ce223b2aace b88154af4513b5a88c23cafe68667cdb3fc225ca91137fdd628db44dba3633a9 9bc795140136289ea74c77be92f985c3441599c4bc1796ec36fc5a3bc54fb319 27201ba4b44fcc0e893ad1360e975ec870dd2b3029e242eaf90feb539637d10e 097584a799611e56adca73f117ab0de4d28e39ca4266d1dc5070b752a26b16a5 c0b05ac4b1848f721810457655081d38a270c286e74c59996f0ed5c7eede2bd4 990f1ac15f200f79449d034395f95b836ea1d779c352f6b76e203c1118f539d2 fa5473b48afb131773601f7124680e95760aac27e14ce6f4bc5dd1be49d5c989 ae52a3c321b116095fe8601d61efb642742925da8efad90fb19a4d6e82a83755 f45b584e79d9e681e9101fec053b761cf1933faa75c94242c7c07ee168a7a70e 28243d60691f391ca3406c62cf5cbfca2fe277f3fd9c9636479a6ee43fb86e43 2e2eca423a43852302d4191893b9aef1511545413856e33bd28fd89fd60b2c5d 67893c342606d93148c7683be4f7192526441f4c4b2850b32946c0add5 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 161 /Gamma put dup 163 /Theta put dup 164 /Lambda put dup 167 /Sigma put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 188 /germandbls put dup 190 /oe put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 52 /four put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 73 /I put dup 75 /K 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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 161 /Gamma put dup 163 /Theta put dup 164 /Lambda put dup 167 /Sigma put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 188 /germandbls put dup 190 /oe put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T 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 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 119 /w put dup 120 /x put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueXX 5000769 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be79011d1f5bfae5c1f476ee6f05eb1d2caeb 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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 161 /Gamma put dup 163 /Theta put dup 164 /Lambda put dup 167 /Sigma put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 188 /germandbls put dup 190 /oe put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 34 /quotedblright put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 196 /dieresis put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueXX 5000793 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306457c7e62c1d269f306bd3402e266de fc3b5e7d8a8d2f5bf0fe6ddd40d07391df4fad4a6018dce29a2b8f692b29f202 3a7c0e66de8ed85c14f1f8492167357f51a7e84cc5d92e0fee4d81cf7fbc8de5 2d2e7bb57142033993f9c08c315abade8dbc4a732e84e142d3bee51557910e12 cd8aa37c459a5e6b7f5269f59078aba3be4641a11ac48d0b625c8325b38ec08e 4c9e5e7fed976a5650d99d82114f449b9ca14c4ec957702295a39a93ef93f618 99b8ea06b092c3c1e503e6e436e0a9fa22576c8930ab3dc8c20f5d82b69cddf8 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CMCSC10 %!PS-AdobeFont-1.1: CMCSC10 1.0 %%CreationDate: 1991 Aug 18 17:46:49 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMCSC10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMCSC10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 163 /Theta put dup 164 /Lambda put dup 167 /Sigma put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /arrowup put dup 175 /arrowdown put dup 176 /quotesingle put dup 177 /exclamdown put dup 178 /questiondown put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 188 /germandbls put dup 190 /oe put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 45 /hyphen put dup 46 /period put dup 64 /at put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 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 dup 196 /dieresis put readonly def /FontBBox{14 -250 1077 750}readonly def /UniqueXX 5000772 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 66aad332fe6fbcced1417802822a3b81d6187875263d6bbda04bbcf6e4870fee ad60f104bb3c5a766610dd11aea64a6b107b0b04439fa2888b8cc39232bb83f7 695aba81f0260cd5248b9e649cd800bcb325468c79ffacb9cd3037a112066957 7889d67fff57549064b985d67ede2fb03272083723541051ebe936eff01a4ab3 cd845d565dfdf25d9ddd2792db17f52f894bea484480f06c832140fdd6c3af0e fef0584c6b78b5e2bbb9a752f10581fd1c48cb498a822afb18cddebba34695b9 b418010368fdfb7bf758d50f3af03360b81c776325c33c9c81130b1e1be840cf 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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 161 /Gamma put dup 163 /Theta put dup 164 /Lambda put dup 167 /Sigma put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 188 /germandbls put dup 190 /oe put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress 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 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 83 /S put dup 84 /T put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 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 119 /w put dup 120 /x put dup 196 /dieresis put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueXX 5000768 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306462c563d51ece66fcc9c831843ecae 1fefc1a232e2724f7baee428ae03aadb95c3035345c15e9922fe49e1f2cfc980 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont TeXDict begin 39158280 55380996 1000 8000 8000 (sdp_icml_final2.dvi) @start /Fa 147[284 6[455 2[569 512 12[740 34[512 2[284 46[{}7 996.264 /CMSL9 rf /Fb 150[295 105[{}1 553.48 /CMMI5 rf /Fc 252[424 2[640{}2 664.176 /CMSY6 rf /Fd 175[1081 60[754 754 18[{}3 996.264 /CMEX10 rf /Fe 134[540 540 740 540 569 398 404 401 1[569 512 569 853 284 540 1[284 569 512 313 455 569 455 569 512 10[768 1[740 569 5[938 7[697 1[740 1[768 3[796 2[284 7[512 512 512 1[284 341 284 796 1[398 398 27[569 12[{}42 996.264 /CMR9 rf /Ff 150[296 4[427 15[486 84[{}3 664.176 /CMMI6 rf /Fg 173[720 82[{}1 996.264 /MSBM10 rf /Fh 201[0 3[683 8[1024 9[796 7[796 1[796 2[796 17[796{}8 996.264 /CMSY9 rf /Fi 134[504 580 20[443 10[846 4[626 7[867 7[731 775 768 5[284 284 35[617 597 21[{}13 996.264 /CMMI9 rf /Fj 204[406 406 406 49[{}3 664.176 /CMR6 rf /Fk 173[799 82[{}1 1106.96 /MSBM10 rf /Fl 148[454 56[597 1[255 24[692 1[692 692 16[454 692 1[692{}9 774.872 /CMSY7 rf /Fm 136[722 119[{}1 774.872 /CMBX7 rf /Fn 140[1107 1107 1[1107 23[1599 7[1168 12[969 969 969 969 3[984 1[984 1[984 6[969 969 4[892 892 8[876 876 12[815 815 4[615 1[523 523 646 646 8[{}25 1106.96 /CMEX10 rf /Fo 194[680 9[441 441 441 441 4[680 1[346 346 40[{}8 774.872 /CMR7 rf /Fp 135[502 640 2[334 1[411 1[456 1[547 786 280 1[366 313 1[432 432 1[461 1[389 13[523 5[701 2[744 3[689 561 3[669 666 5[263 33[518 1[444 538 525 7[465 502 575 11[{}29 774.872 /CMMI7 rf /Fq 133[515 543 633 792 537 1[400 5[664 972 1[576 456 381 1[528 542 515 576 479 475 585 7[643 917 4[679 841 2[844 3[940 1[487 1[870 712 2[791 840 830 2[861 553 861 307 307 27[660 2[633 1[631 1[547 667 646 5[449 492 573 626 708 11[{}46 1106.96 /CMMI10 rf /Fr 143[922 4[553 307 430 430 553 553 12[740 790 7[770 3[763 843 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e(Computer)i(Science,)f(U.C.)g(Berk)-31 b(eley)-1840 13381 y Fw(Nello)425 b(Cristianini)29623 b Fv(nello@suppor)-74 b(t-vector.net)-1840 14709 y Fw(P)-35 b(eter)425 b(Bartlett)30495 b Fv(peter.bar)-74 b(tlett@anu.edu.a)-25 b(u)-1840 16278 y Fu(BIOwulf)370 b(T)-92 b(ec)-31 b(hnologies)371 b(/)f(Division)g(of)g (Computer)g(Science)g(and)f(Departmen)-31 b(t)370 b(of)g(Statistics,)i (U.C.)e(Berk)-31 b(eley)-1840 18406 y Fw(Lauren)c(t)425 b(El)f(Ghaoui)27488 b Fv(elgha)-25 b(oui@eecs.berkeley.edu)-1840 19974 y Fu(Departmen)-31 b(t)370 b(of)g(Electrical)h(Engineering)g(and) e(Computer)i(Science,)f(U.C.)g(Berk)-31 b(eley)-1840 22103 y Fw(Mic)c(hael)424 b(I.)g(Jordan)30341 b Fv(jord)-25 b(an@cs.berkeley.edu)-1840 23671 y Fu(Division)370 b(of)g(Computer)h (Science)e(and)g(Departmen)-31 b(t)371 b(of)e(Statistics,)j(U.C.)f (Berk)-31 b(eley)8304 28193 y Ft(Abstract)374 30530 y Fu(Kernel-based)698 b(learning)h(algorithms)h(w)-31 b(ork)699 b(b)-31 b(y)374 31858 y(em)g(b)31 b(edding)431 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y(the)591 b(geometry)h(of)g(the)f(em)-31 b(b)31 b(edding)592 b(space)f(and)374 46470 y(inducing)477 b(a)f(notion)h(of)g(similarit)-31 b(y)478 b(in)e(the)g(input)374 47798 y(space|classical)413 b(mo)31 b(del)412 b(selection)g(problems)g(in)374 49127 y(mac)-31 b(hine)307 b(learning.)473 b(In)305 b(this)i(pap)31 b(er)305 b(w)-31 b(e)306 b(sho)-31 b(w)307 b(ho)-31 b(w)374 50455 y(the)458 b(k)-31 b(ernel)458 b(matrix)h(can)f(b)31 b(e)457 b(learned)h(from)h(data)374 51784 y(via)556 b(Semi-De\257nite)h (Programming)h(tec)-31 b(hniques.)374 53112 y(When)440 b(applied)i(to)g(a)f(k)-31 b(ernel)440 b(matrix)j(asso)31 b(ciated)374 54440 y(with)370 b(b)31 b(oth)369 b(training)h(and)f(test) g(data)h(this)f(giv)-31 b(es)370 b(a)374 55769 y(p)31 b(o)-31 b(w)g(erful)543 b(transductiv)-31 b(e)543 b(algorithm|using)j (the)374 57097 y(lab)31 b(elled)545 b(part)f(of)g(the)g(data)g(one)g (can)g(learn)g(an)374 58425 y(\\optimal")381 b(em)-31 b(b)31 b(edding)378 b(also)f(for)g(the)h(unlab)31 b(elled)374 59754 y(part.)1072 b(The)563 b(induced)f(similarit)-31 b(y)565 b(b)31 b(et)-31 b(w)g(een)564 b(test)374 61082 y(p)31 b(oin)-31 b(ts)650 b(is)e(learned)h(b)-31 b(y)649 b(using)h(training)g(p)31 b(oin)-31 b(ts)374 62410 y(and)382 b(their)f(lab)31 b(els.)530 b(Imp)31 b(ortan)-31 b(tly)-92 b(,)386 b(these)c(learning)374 63739 y(problems)374 b(are)f(con)-31 b(v)g(ex,)376 b(so)d(w)-31 b(e)374 b(obtain)g(a)g(metho)31 b(d)374 65067 y(for)589 b(learning)h(b)31 b(oth)589 b(the)g(mo)31 b(del)589 b(class)g(and)g(the)374 66395 y(function)447 b(without)h(lo)31 b(cal)447 b(minima.)725 b(Finally)-92 b(,)467 b(the)374 67724 y(no)-31 b(v)g(el)303 b(approac)-31 b(h)303 b(presen)-31 b(ted)301 b(in)h(the)g(pap)31 b(er)302 b(is)g(sup-)374 69052 y(p)31 b(orted)369 b(b)-31 b(y)370 b(p)31 b(ositiv)-31 b(e)370 b(empirical)h(results.)26160 28193 y Ft(1.)498 b(In)-42 b(tro)42 b(duction)26160 30346 y Fu(Finding)299 b(a)g(represen)-31 b(tation)299 b(of)g(the)g(data)g (that)h(mak)-31 b(es)299 b(it)g(eas-)26160 31674 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b(w)g(e-T)-92 b(a)-31 b(ylor,)506 b(2000;)532 b(Sc)-31 b(h\304)-553 b(olk)-31 b(opf)479 b(&)d(Smola,)505 b(2002\))478 b(w)-31 b(ork)-1840 38213 y(b)g(y)661 b(em)-31 b(b)31 b(edding)662 b(the)f(data)h(in)-31 b(to)663 b(a)e(Hilb)31 b(ert)661 b(space,)735 b(and)-1840 39542 y(searc)-31 b(hing)559 b(for)e(linear)i(relations)g(in)f(suc)-31 b(h)558 b(a)g(space.)1059 b(The)-1840 40870 y(em)-31 b(b)31 b(edding)469 b(is)e(p)31 b(erformed)467 b(implicitly)-92 b(,)496 b(b)-31 b(y)467 b(sp)31 b(ecifying)469 b(the)-1840 42198 y(inner)349 b(pro)31 b(duct)350 b(b)31 b(et)-31 b(w)g(een)350 b(eac)-31 b(h)350 b(pair)g(of)g(p)31 b(oin)-31 b(ts)350 b(rather)f(than)-1840 43527 y(b)-31 b(y)442 b(giving)i(their)e(co)31 b(ordinates)442 b(explicitly)-92 b(.)713 b(This)442 b(approac)-31 b(h)-1840 44855 y(has)475 b(sev)-31 b(eral)475 b(adv)-61 b(an)-31 b(tages,)503 b(the)475 b(most)h(imp)31 b(ortan)-31 b(t)477 b(deriving)-1840 46183 y(from)392 b(the)g(fact)g(that)h(often)g(the)e(inner)h(pro)31 b(duct)391 b(in)h(the)g(em-)-1840 47512 y(b)31 b(edding)270 b(space)f(can)h(b)31 b(e)269 b(computed)i(m)-31 b(uc)g(h)270 b(more)g(easily)g(than)-1840 48840 y(the)369 b(co)31 b(ordinates)371 b(of)e(the)h(p)31 b(oin)-31 b(ts)370 b(themselv)-31 b(es.)-1840 50833 y(Giv)g(en)531 b(an)g(input)g(set)f Fr(X)162 b Fu(,)571 b(and)530 b(an)h(em)-31 b(b)31 b(edding)532 b(space)e Fr(F)110 b Fu(,)-1840 52161 y(w)-31 b(e)518 b(consider)f(a)g(map)h(\251)555 b(:)f Fr(X)716 b(!)554 b(F)110 b Fu(.)936 b(Giv)-31 b(en)518 b(t)-31 b(w)g(o)519 b(p)31 b(oin)-31 b(ts)-1840 53489 y Fq(x)-1207 53655 y Fp(i)-281 53489 y Fr(2)558 b(X)680 b Fu(and)520 b Fq(x)5421 53655 y Fp(j)6445 53489 y Fr(2)558 b(X)162 b Fu(,)556 b(the)520 b(function)h(that)f(returns)f(the)-1840 54818 y(inner)400 b(pro)31 b(duct)401 b(b)31 b(et)-31 b(w)g(een)402 b(their)f(images)h(in)f(the)f(space)h Fr(F)510 b Fu(is)-1840 56146 y(kno)-31 b(wn)370 b(as)g(the)f Fs(kernel)397 b(function)p Fu(.)-1840 58139 y Fw(De\257nition)790 b Fu(A)f Fs(kernel)783 b 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b(matrix")g(or)f(\\Gram)g (ma-)-1840 70571 y(trix.")653 b(It)422 b(is)g(a)g(symmetric)i(p)31 b(ositiv)-31 b(e)423 b(de\257nite)g(matrix,)438 b(and)26160 -322 y(since)420 b(it)h(sp)31 b(eci\257es)419 b(the)i(inner)f(pro)31 b(ducts)419 b(b)31 b(et)-31 b(w)g(een)421 b(all)h(pairs)26160 1006 y(of)480 b(p)31 b(oin)-31 b(ts)480 b Fr(f)p Fq(x)32154 1172 y Fp(i)32523 1006 y Fr(g)33076 605 y Fp(n)33076 1295 y(i)p Fo(=1)34566 1006 y Fu(,)507 b(it)480 b(completely)h (determines)f(the)g(rela-)26160 2335 y(tiv)-31 b(e)409 b(p)31 b(ositions)409 b(b)31 b(et)-31 b(w)g(een)408 b(those)g(p)31 b(oin)-31 b(ts)409 b(in)f(the)g(em)-31 b(b)31 b(edding)26160 3663 y(space.)26160 5656 y(Since)406 b(in)h(this)f(pap)31 b(er)406 b(w)-31 b(e)407 b(will)g(consider)f(a)h Fs(\257nite)490 b Fu(input)407 b(set)26160 6984 y Fr(X)162 b Fu(,)426 b(w)-31 b(e)416 b(can)f(c)-31 b(haracterize)416 b(k)-31 b(ernel)416 b(functions)g(and)f(matrices)26160 8312 y(in)369 b(the)h(follo)-31 b(wing)373 b(simple)d(w)-31 b(a)g(y)-92 b(.)26160 10305 y Fw(Prop)35 b(osition)528 b Fu(Ev)-31 b(ery)526 b(p)31 b(ositiv)-31 b(e)527 b(de\257nite)f(and)g(symmetric) 26160 11633 y(matrix)499 b(is)f(a)f(k)-31 b(ernel)498 b(matrix,)532 b(that)499 b(is,)530 b(an)498 b(inner)f(pro)31 b(duct)26160 12962 y(matrix)489 b(in)e(some)h(em)-31 b(b)31 b(edding)488 b(space.)847 b(Con)-31 b(v)g(ersely)-92 b(,)518 b(ev)-31 b(ery)26160 14290 y(k)g(ernel)369 b(matrix)i(is)e (symmetric)i(and)e(p)31 b(ositiv)-31 b(e)371 b(de\257nite.)26160 16283 y(Notice)293 b(that,)309 b(if)293 b(w)-31 b(e)292 b(ha)-31 b(v)g(e)293 b(a)f(k)-31 b(ernel)293 b(matrix,)309 b(w)-31 b(e)292 b(do)g(not)h(need)26160 17611 y(to)328 b(kno)-31 b(w)328 b(the)g(k)-31 b(ernel)327 b(function,)338 b(nor)327 b(the)h(implicitly)i(de\257ned)26160 18939 y(map)442 b(\251,)460 b(nor)441 b(the)h(co)31 b(ordinates)442 b(of)g(the)g(p)31 b(oin)-31 b(ts)442 b(\251\()p Fq(x)48807 19105 y Fp(i)49176 18939 y Fu(\).)710 b(W)-92 b(e)26160 20268 y(do)302 b(not)g(ev)-31 b(en)302 b(need)f Fr(X)463 b Fu(to)302 b(b)31 b(e)301 b(a)h(v)-31 b(ector)302 b(space;)324 b(in)302 b(fact)h(in)e(this)26160 21596 y(pap)31 b(er)309 b(it)i(will)g(b)31 b(e)310 b(a)g(generic)g(\257nite)g(set.)473 b(W)-92 b(e)309 b(are)g(guaran)-31 b(teed)26160 22924 y(that)444 b(the)f(data)h(are)f(implicitly)j(mapp)31 b(ed)443 b(to)h(some)f(Hilb)31 b(ert)26160 24253 y(space)344 b(b)-31 b(y)345 b(simply)h(c)-31 b(hec)g(king)345 b(that)h(the)f(k)-31 b(ernel)344 b(matrix)i(satis-)26160 25581 y(\257es)369 b(the)g(conditions)i(ab)31 b(o)-31 b(v)g(e.)26160 27574 y(The)381 b(solutions)g(sough)-31 b(t)382 b(b)-31 b(y)380 b(k)-31 b(ernel-based)381 b(algorithms)h(suc)-31 b(h)26160 28902 y(as)361 b(the)f(supp)31 b(ort)361 b(v)-31 b(ector)361 b(mac)-31 b(hine)362 b(\(SVM\))f(are)g(linear)g(func-)26160 30230 y(tions)370 b(in)g(the)f(feature)h(space:)35188 32561 y Fq(f)119 b Fu(\()p Fw(x)p Fu(\))308 b(=)g Fw(w)39795 32005 y Fp(T)40492 32561 y Fu(\251\()p Fw(x)p Fu(\))p Fq(;)26160 34892 y Fu(for)h(some)g(w)-31 b(eigh)g(t)311 b(v)-31 b(ector)309 b Fw(w)18 b Fu(.)472 b(The)309 b(k)-31 b(ernel)309 b(can)g(b)31 b(e)308 b(exploited)26160 36220 y(whenev)-31 b(er)248 b(the)h(w)-31 b(eigh)g(t)250 b(v)-31 b(ector)249 b(can)f(b)31 b(e)248 b(expressed)f(as)h(a)g(linear)26160 37549 y(com)-31 b(bination)267 b(of)d(the)g(training)h(p)31 b(oin)-31 b(ts,)286 b Fw(w)325 b Fu(=)45233 36719 y Fn(P)46401 36992 y Fp(n)46401 37881 y(i)p Fo(=1)48075 37549 y Fq(\256)48783 37715 y Fp(i)49152 37549 y Fu(\251\()p Fw(x)51053 37715 y Fp(i)51422 37549 y Fu(\),)26160 38877 y(implying)372 b(that)e(w)-31 b(e)370 b(can)f(express)f Fq(f)488 b Fu(as)369 b(follo)-31 b(ws:)33965 42034 y Fq(f)119 b Fu(\()p Fw(x)p Fu(\))308 b(=)38037 40650 y Fp(n)37512 40982 y Fn(X)37594 43340 y Fp(i)p Fo(=1)39295 42034 y Fq(\256)40003 42200 y Fp(i)40372 42034 y Fq(k)35 b Fu(\()p Fw(x)42085 42200 y Fp(i)42454 42034 y Fq(;)184 b Fw(x)p Fu(\))p Fq(:)26160 45997 y Fu(An)363 b(imp)31 b(ortan)-31 b(t)366 b(issue)d(in)g (applications)j(is)d(that)i(of)f(c)-31 b(ho)31 b(osing)26160 47325 y(a)275 b(k)-31 b(ernel)275 b Fq(k)310 b Fu(for)275 b(a)h(giv)-31 b(en)276 b(learning)g(task;)307 b(in)-31 b(tuitiv)g(ely)-92 b(,)298 b(w)-31 b(e)275 b(wish)26160 48653 y(to)431 b(c)-31 b(ho)31 b(ose)431 b(a)f(k)-31 b(ernel)430 b(that)i(induces)e(the)g(\\righ)-31 b(t")433 b(metric)e(in)26160 49982 y(the)369 b(space.)26160 51974 y(F)-92 b(or)392 b(the)g(sp)31 b(ecial)393 b(case)f(of)h(t)-31 b(w)g(o-class)394 b(classi\257cation,)401 b(sev)-31 b(eral)26160 53303 y(measures)385 b(of)i(separation)g(b)31 b(et)-31 b(w)g(een)387 b(t)-31 b(w)g(o)388 b(sets)d(of)i(data)g(ha)-31 b(v)g(e)26160 54631 y(b)31 b(een)328 b(dev)-31 b(elop)31 b(ed.)480 b(F)-92 b(or)328 b(example,)339 b(w)-31 b(e)330 b(can)f(de\257ne)f(the)h Fs(align-)26160 55959 y(ment)341 b Fu(b)31 b(et)-31 b(w)g(een)341 b(a)g(k)-31 b(ernel)340 b(and)h(a)f(set)g(of)h(lab)31 b(els,)348 b(or)340 b(the)g Fs(mar-)26160 57288 y(gin)436 b Fu(of)g(a)g(separation)g(\(the)h 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67914 y(Giv)-31 b(en)249 b(an)h(\(unlab)31 b(elled\))251 b(sample)f Fq(S)371 b Fu(=)307 b Fr(f)p Fw(x)43742 68080 y Fo(1)44239 67914 y Fq(;)184 b(:)g(:)g(:)k(;)c Fw(x)47370 68080 y Fp(n)47973 67914 y Fr(g)p Fu(,)274 b(w)-31 b(e)249 b(use)26160 69243 y(the)273 b(follo)-31 b(wing)277 b(\(F)-92 b(rob)31 b(enius\))273 b(inner)f(pro)31 b(duct)273 b(b)31 b(et)-31 b(w)g(een)274 b(Gram)26160 70571 y(matrices,)371 b Fr(h)p Fq(K)32275 70737 y Fo(1)32772 70571 y Fq(;)184 b(K)34203 70737 y Fo(2)34701 70571 y Fr(i)35131 70737 y Fp(F)36174 70571 y Fu(=)37343 69741 y Fn(P)38511 70015 y Fp(n)38511 70903 y(i;j)45 b Fo(=1)40859 70571 y Fq(k)41435 70737 y Fo(1)41932 70571 y Fu(\()p Fw(x)43034 70737 y Fp(i)43403 70571 y Fq(;)184 b Fw(x)44566 70737 y Fp(j)45033 70571 y Fu(\))p Fq(k)46039 70737 y Fo(2)46537 70571 y Fu(\()p Fw(x)47639 70737 y Fp(i)48007 70571 y Fq(;)g Fw(x)49170 70737 y Fp(j)49638 70571 y Fu(\).)p eop %%Page: 3 3 3 2 bop -1840 -322 a Fw(Alignmen)-35 b(t)401 b Fu(The)f(\(empirical\))j (alignmen)-31 b(t)403 b(of)d(a)h(k)-31 b(ernel)400 b Fq(k)23663 -156 y Fo(1)-1840 1006 y Fu(with)445 b(a)f(k)-31 b(ernel)444 b Fq(k)5666 1172 y Fo(2)6607 1006 y Fu(with)h(resp)31 b(ect)443 b(to)i(the)f(sample)h Fq(S)507 b Fu(is)444 b(the)-1840 2335 y(quan)-31 b(tit)g(y)2161 4858 y(^)1869 5138 y Fq(A)p Fu(\()p Fq(S)r(;)184 b(k)4877 5304 y Fo(1)5375 5138 y Fq(;)g(k)6442 5304 y Fo(2)6940 5138 y Fu(\))308 b(=)12014 4389 y Fr(h)p Fq(K)13384 4555 y Fo(1)13881 4389 y Fq(;)184 b(K)15312 4555 y Fo(2)15810 4389 y Fr(i)16240 4555 y Fp(F)p 8979 4883 11032 45 v 8979 5104 a Fn(p)p 10086 5104 9925 45 v 947 x Fr(h)p Fq(K)11456 6217 y Fo(1)11954 6051 y Fq(;)g(K)13385 6217 y Fo(1)13882 6051 y Fr(i)14312 6217 y Fp(F)15049 6051 y Fr(h)p Fq(K)16419 6217 y Fo(2)16916 6051 y Fq(;)g(K)18347 6217 y Fo(2)18845 6051 y Fr(i)19275 6217 y Fp(F)20144 5138 y Fq(;)-1840 8166 y Fu(where)399 b Fq(K)2331 8332 y Fp(i)3099 8166 y Fu(is)h(the)f(k)-31 b(ernel)400 b(matrix)h(for)f(the)f(sample)i Fq(S)462 b Fu(using)-1840 9494 y(k)-31 b(ernel)369 b Fq(k)1998 9660 y Fp(i)2367 9494 y Fu(.)-1840 11487 y(This)g(can)g(also)g(b)31 b(e)368 b(view)-31 b(ed)370 b(as)e(the)h(cosine)g(of)g(the)g(angle)g(b) 31 b(e-)-1840 12815 y(t)-31 b(w)g(een)470 b(t)-31 b(w)g(o)470 b(bi-dimensional)h(v)-31 b(ectors)469 b Fq(K)15871 12981 y Fo(1)16836 12815 y Fu(and)g Fq(K)20028 12981 y Fo(2)20524 12815 y Fu(,)494 b(repre-)-1840 14143 y(sen)-31 b(ting)383 b(the)f(Gram)g(matrices.)532 b(If)382 b(w)-31 b(e)382 b(consider)g Fq(K)19975 14309 y Fo(2)20801 14143 y Fu(=)328 b Fq(y)40 b(y)23156 13742 y Fp(T)23853 14143 y Fu(,)-1840 15472 y(where)470 b Fq(y)509 b Fu(is)469 b(the)h(v)-31 b(ector)471 b(of)f Fr(f\247)p Fu(1)p Fr(g)h Fu(lab)31 b(els)471 b(for)e(the)i(sample,)-1840 16800 y(then)-1548 19344 y(^)-1840 19624 y Fq(A)p Fu(\()p Fq(S)r(;)184 b(K)18 b(;)184 b(y)40 b(y)3207 19167 y Fp(T)3906 19624 y Fu(\))308 b(=)8984 18875 y Fr(h)p Fq(K)18 b(;)184 b(y)40 b(y)12029 18474 y Fp(T)12726 18875 y Fr(i)13156 19041 y Fp(F)p 5945 19369 10987 45 v 5945 19591 a Fn(p)p 7052 19591 9880 45 v 956 x Fr(h)p Fq(K)18 b(;)184 b(K)79 b Fr(i)10380 20713 y Fp(F)11118 20547 y Fr(h)p Fq(y)40 b(y)12714 20227 y Fp(T)13411 20547 y Fq(;)184 b(y)40 b(y)15068 20227 y Fp(T)15765 20547 y Fr(i)16195 20713 y Fp(F)17372 19624 y Fu(=)19199 17853 y Fn(\255)19721 18750 y Fq(K)18 b(;)184 b(y)40 b(y)22336 18348 y Fp(T)23033 17853 y Fn(\256)23556 19192 y Fp(F)p 18673 19369 6145 45 v 18673 20537 a Fq(m)19645 19591 y Fn(p)p 20752 19591 4067 45 v 946 x Fr(h)p Fq(K)18 b(;)184 b(K)79 b Fr(i)24080 20703 y Fp(F)24951 19624 y Fq(;)-1840 22824 y Fu(since)872 21928 y Fn(\255)1395 22824 y Fq(y)40 b(y)2561 22423 y Fp(T)3257 22824 y Fq(;)184 b(y)40 b(y)4914 22423 y Fp(T)5611 21928 y Fn(\256)6134 23267 y Fp(F)7177 22824 y Fu(=)308 b Fq(m)9318 22423 y Fo(2)9814 22824 y Fu(.)-1840 24817 y(Secondly)-92 b(,)716 b(w)-31 b(e)646 b(will)h(de\257ne)e(the)h(margin)h(of)f(separation,) -1840 26145 y(whic)-31 b(h)370 b(will)h(then)f(b)31 b(e)368 b(expressed)g(using)i(dualit)-31 b(y)-92 b(.)-1840 28138 y Fw(Margin)570 b Fu(Giv)-31 b(en)570 b(a)f(linearly)i(separable)e(lab) 31 b(elled)571 b(sample)-1840 29466 y Fq(S)-1161 29632 y Fp(l)-118 29466 y Fu(=)704 b Fr(f)p Fu(\()p Fw(x)3102 29632 y Fo(1)3600 29466 y Fq(;)184 b(y)4634 29632 y Fo(1)5131 29466 y Fu(\))p Fq(;)g(:)g(:)g(:)k(;)c Fu(\()p Fw(x)9122 29632 y Fp(n)9726 29466 y Fq(;)g(y)10760 29632 y Fp(n)11364 29466 y Fu(\))p Fr(g)p Fu(,)667 b(the)607 b(h)-31 b(yp)31 b(erplane)608 b(\()p Fw(w)18 b Fq(;)184 b(b)p Fu(\))-1840 30795 y(that)370 b(solv)-31 b(es)370 b(the)g(optimization)j(problem) 2301 33021 y(min)2530 33743 y Fm(w)14 b Fp(;b)6360 33021 y Fr(h)p Fw(w)k Fq(;)184 b Fw(w)18 b Fr(i)13161 b Fu(\(1\))-688 35278 y(sub)61 b(ject)370 b(to)2215 b Fq(y)6903 35444 y Fp(i)7271 35278 y Fu(\()p Fr(h)p Fw(w)18 b Fq(;)184 b Fu(\251\()p Fw(x)11460 35444 y Fp(i)11832 35278 y Fu(\))p Fr(i)247 b Fu(+)f Fq(b)p Fu(\))308 b Fr(\270)f Fu(1)p Fq(;)184 b(i)309 b Fu(=)f(1)p Fq(;)184 b(:)g(:)g(:)k(;)c(n)-1840 37504 y Fu(realizes)563 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b(2000\).)480 b(By)323 b(transforming)h(\(1\)) g(in)-31 b(to)-1840 50123 y(its)378 b(corresp)31 b(onding)377 b(dual)h(problem)g(\(Cristianini)i(&)d(Sha)-31 b(w)g(e-)-1840 51452 y(T)-92 b(a)-31 b(ylor,)311 b(2000\),)g(w)-31 b(e)294 b(can)f(express)f(the)i(squared)f(in)-31 b(v)g(erse)294 b(mar-)-1840 52780 y(gin)379 b Fq(w)30 b Fu(\()p Fq(K)79 b Fu(\))325 b(=)e(1)p Fq(=\260)5964 52378 y Fo(2)6840 52780 y Fu(corresp)31 b(onding)379 b(to)g(a)g(k)-31 b(ernel)379 b(matrix)h Fq(K)-1840 54108 y Fu(as)369 b(follo)-31 b(ws:)-1840 56334 y Fq(w)30 b Fu(\()p Fq(K)79 b Fu(\))1109 b(=)e Fr(h)p Fw(w)5287 56500 y Fl(\244)5796 56334 y Fq(;)184 b Fw(w)7206 56500 y Fl(\244)7717 56334 y Fr(i)15470 b Fu(\(2\))1970 57995 y(=)1107 b(max)4680 58659 y Fp(\256)6490 57995 y Fu(2)p Fq(\256)7755 57538 y Fp(T)8453 57995 y Fq(e)246 b Fr(\241)g Fq(\256)11033 57538 y Fp(T)11731 57995 y Fq(G)p Fu(\()p Fq(K)79 b Fu(\))p Fq(\256)682 b Fu(:)677 b Fq(\256)312 b Fr(\270)307 b Fu(0)p Fq(;)801 b(\256)21415 57538 y Fp(T)22112 57995 y Fq(y)347 b Fu(=)307 b(0)p Fq(;)-1840 60609 y Fu(where)563 b Fq(e)h Fu(is)g(the)g Fq(n)p Fu(-v)-31 b(ector)565 b(of)f(ones,)613 b Fq(\256)636 b Fr(2)631 b Fk(R)18579 60207 y Fp(n)19182 60609 y Fu(,)613 b Fq(G)p Fu(\()p Fq(K)79 b Fu(\))565 b(is)-1840 61937 y(de\257ned)536 b(b)-31 b(y)537 b Fq(G)4715 62103 y Fp(ij)5495 61937 y Fu(\()p Fq(K)79 b Fu(\))588 b(=)e([)p Fq(K)79 b Fu(])11042 62103 y Fp(ij)11823 61937 y Fq(y)12366 62103 y Fp(i)12734 61937 y Fq(y)13277 62103 y Fp(j)14330 61937 y Fu(=)586 b Fq(k)35 b Fu(\()p Fw(x)17490 62103 y Fp(i)17859 61937 y Fq(;)184 b Fw(x)19022 62103 y Fp(j)19489 61937 y Fu(\))p Fq(y)20462 62103 y Fp(i)20831 61937 y Fq(y)21374 62103 y Fp(j)22377 61937 y Fu(and)-1840 63265 y Fq(\256)312 b Fr(\270)307 b Fu(0)h Fr(,)g Fq(\256)3332 63431 y Fp(i)4008 63265 y Fr(\270)f Fu(0)p Fq(;)555 b(i)307 b Fu(=)h(1)p Fq(;)184 b(:)g(:)g(:)k(;)c(n)p Fu(.)-1840 65258 y(F)-92 b(or)556 b(a)h(non-linearly)i(separable)e(lab)31 b(elled)558 b(sample)g Fq(S)21649 65424 y Fp(l)21988 65258 y Fu(,)604 b(w)-31 b(e)-1840 66586 y(can)538 b(de\257ne)g(the)g(noise-toleran)-31 b(t)541 b(soft)e(margin)g(\(Sc)-31 b(h\304)-553 b(olk)-31 b(opf)-1840 67914 y(&)504 b(Smola,)541 b(2002\).)901 b(Geometrically)-92 b(,)541 b(this)505 b(corresp)31 b(onds)503 b(to)-1840 69243 y(the)400 b(distance)h(b)31 b(et)-31 b(w)g(een)401 b(reduced)f(con)-31 b(v)g(ex)401 b(h)-31 b(ulls)401 b(of)g(the)f(t)-31 b(w)g(o)-1840 70571 y(classes)369 b(\(Bennett)h(&)f(Bredensteiner,)g(2000\).)26160 -322 y(Both)425 b(the)g(alignmen)-31 b(t)427 b(\(Cristianini)g(et)d(al.,)440 b(2001\))426 b(and)f(the)26160 1006 y(soft)557 b(margin)h(\(Sc)-31 b(h\304)-553 b(olk)-31 b(opf)559 b(&)d(Smola,)606 b(2002\))558 b(are)f(concen-)26160 2335 y(trated)543 b(quan)-31 b(tities.)1016 b(This)543 b(means)g(that)g(w)-31 b(e)543 b(can)g(reliably)26160 3663 y(estimate)382 b(their)e(exp)31 b(ected)380 b(v)-61 b(alue)380 b(from)g(a)h(\257nite)f(sample,)k(or)26160 4992 y(estimate)346 b(their)e(v)-61 b(alue)344 b(on)g(the)g(test)g(set) g(b)-31 b(y)344 b(observing)g(them)26160 6320 y(on)369 b(the)h(training)h(set.)26160 8312 y(These)499 b(quan)-31 b(tities)501 b(can)e(no)-31 b(w)500 b(b)31 b(e)499 b(used)f(to)i (obtain)g(b)31 b(ounds)26160 9641 y(on)321 b(the)h(generalization)i(of) d(giv)-31 b(en)322 b(classi\257ers)f(\(under)g(the)g(as-)26160 10969 y(sumption)484 b(that)h(the)e(data)i(ha)-31 b(v)g(e)484 b(b)31 b(een)482 b(dra)-31 b(wn)484 b(i.i.d.\).)837 b(As)26160 12297 y(w)-31 b(e)429 b(will)h(see)d(in)i(Section)g(5,)444 b Fq(\260)490 b Fu(can)428 b(b)31 b(e)428 b(used)f(to)i(b)31 b(ound)429 b(the)26160 13626 y(p)31 b(erformance)333 b(of)h(supp)31 b(ort)333 b(v)-31 b(ector)333 b(mac)-31 b(hines:)476 b(for)333 b(a)g(thresh-)26160 14954 y(olded)365 b(v)-31 b(ersion)364 b(of)g Fq(f)119 b Fu(\()p Fw(x)p Fu(\))309 b(=)e Fr(h)p Fw(w)18 b Fq(;)184 b Fu(\251\()p Fw(x)p Fu(\))p Fr(i)p Fu(,)369 b(with)c Fr(k)p Fw(w)18 b Fr(k)47641 14552 y Fo(2)48446 14954 y Fu(=)307 b(1)p Fq(=\260)51354 14552 y Fo(2)51853 14954 y Fu(,)26160 16283 y(if)497 b Fq(y)27845 16449 y Fp(i)28213 16283 y Fq(f)119 b Fu(\()p Fw(x)29976 16449 y Fp(i)30345 16283 y Fu(\))519 b Fr(\270)g Fu(1)496 b(for)g Fq(i)520 b Fu(=)e(1)p Fq(;)184 b(:)g(:)g(:)189 b(;)184 b(n)p Fu(,)529 b(then)497 b(the)f(prop)31 b(ortion)26160 17611 y(of)475 b(errors)e(on)h(the)h (test)g(data)g(is,)501 b(with)476 b(probabilit)-31 b(y)477 b(1)317 b Fr(\241)f Fq(\261)42 b Fu(,)26160 18939 y(b)31 b(ounded)369 b(b)-31 b(y)31567 21786 y(1)p 31050 22280 1587 45 v 31050 22501 a Fr(p)p 31972 22501 665 45 v 798 x Fq(n)32954 20642 y Fn(\303)33830 22535 y Fu(4)247 b(+)35737 21534 y Fn(p)p 36844 21534 4670 45 v 1001 x Fu(2)184 b(log)18 b(\(1)p Fq(=\261)42 b Fu(\))247 b(+)42866 20549 y Fn(s)p 43973 20549 2062 45 v 44121 21786 a Fq(B)56 b(C)p 44106 22280 1796 45 v 44106 23294 a(n\260)45404 22974 y Fo(2)46035 20642 y Fn(!)47096 22535 y Fq(;)3343 b Fu(\(3\))26160 26055 y(where)284 b(trace\()p Fq(K)79 b Fu(\))309 b Fr(\267)e Fq(B)56 b Fu(,)301 b(and)284 b(the)g(k)-31 b(ernel)284 b(matrix)h Fq(K)363 b Fu(is)283 b(of)i(the)26160 27383 y(form)501 b Fq(K)604 b Fu(=)31839 26553 y Fn(P)33007 26826 y Fp(m)33007 27715 y(i)p Fo(=1)34681 27383 y Fq(\271)35348 27549 y Fp(i)35716 27383 y Fq(K)36656 27549 y Fp(i)37525 27383 y Fu(for)500 b(a)g(\257xed)g(set)g Fr(f)p Fq(K)46561 27549 y Fo(1)47058 27383 y Fq(;)184 b(:)g(:)g(:)k(;)c(K)50457 27549 y Fp(m)51299 27383 y Fr(g)p Fu(,)26160 28711 y(and)514 b Fq(C)593 b Fu(dep)31 b(ends)512 b(on)i(these)f Fq(m)h Fu(k)-31 b(ernel)513 b(matrices.)927 b(As)513 b(w)-31 b(e)26160 30040 y(will)401 b(see)d(in)h(Section)h(4,)407 b(this)399 b(is)g(exactly)i(what)f(w)-31 b(e)399 b(need)g(for)26160 31368 y(learning)370 b(the)g(k)-31 b(ernel)369 b(matrix.)26160 34542 y Ft(3.)498 b(Semi-De\257nite)i (Programming)26160 36695 y Fu(Semi-de\257nite)3004 b(programming)g(\(V) -92 b(anden)-31 b(b)31 b(erghe)26160 38023 y(&)1215 b(Bo)-31 b(yd,)1427 b(1996\))1217 b(deals)e(with)h(the)f(optimization)26160 39351 y(of)1334 b(con)-31 b(v)g(ex)1334 b(functions)g(o)-31 b(v)g(er)1334 b(the)g(con)-31 b(v)g(ex)1334 b(cone)51663 38950 y Fo(1)26160 40680 y Fr(P)1302 b Fu(=)30303 39783 y Fn(\251)30949 40680 y Fq(X)395 b Fr(2)307 b Fk(R)34105 40278 y Fp(p)p Fl(\243)p Fp(p)35765 40680 y Fr(j)p Fq(X)395 b Fu(=)308 b Fq(X)39557 40278 y Fp(T)40254 40680 y Fq(;)184 b(X)395 b Fr(\272)308 b Fu(0)43779 39783 y Fn(\252)45336 40680 y Fu(of)912 b(symmetric)26160 42008 y(p)31 b(ositiv)-31 b(e)316 b(semi-de\257nite)g(matrices,)327 b(or)315 b(subsets)f(of)h (this)g(cone.)26160 43336 y(Giv)-31 b(en)424 b(the)h(Prop)31 b(osition)426 b(in)e(Section)h(2,)439 b Fr(P)515 b Fu(can)424 b(b)31 b(e)424 b(view)-31 b(ed)26160 44665 y(as)494 b(a)h(searc)-31 b(h)494 b(space)h(for)f(p)31 b(ossible)495 b(k)-31 b(ernel)495 b(matrices.)869 b(This)26160 45993 y(consideration)500 b(leads)f(to)h(the)f(k)-31 b(ey)499 b(problem)g(addressed)f(in)26160 47321 y(this)372 b(pap)31 b(er|w)-31 b(e)372 b(wish)g(to)h(sp)31 b(ecify)371 b(a)h(con)-31 b(v)g(ex)373 b(cost)f(function)26160 48650 y(that)377 b(will)h(enable)f(us)f(to)h(learn)f(the)h(optimal)i(k) -31 b(ernel)376 b(matrix)26160 49978 y(from)608 b Fr(P)698 b Fu(using)607 b(semi-de\257nite)h(programming.)1209 b(Because)26160 51306 y(of)630 b(the)g(con)-31 b(v)g(exit)g(y)-92 b(,)697 b(this)630 b(approac)-31 b(h)631 b(allo)-31 b(ws)632 b(us)d(to)h(a)-31 b(v)g(oid)26160 52635 y(problems)370 b(with)g(lo)31 b(cal)371 b(minima.)26160 54627 y Fw(De\257nition)404 b Fu(The)g(general)f(form)h(of)g(a)f(Semi-De\257nite)i(Pro-)26160 55956 y(gram)370 b(\(SDP\))g(is)29970 58479 y(min)30641 59144 y Fp(x)34029 58479 y Fq(c)34508 58022 y Fp(T)35205 58479 y Fq(x)14908 b Fu(\(4\))26981 60527 y(sub)61 b(ject)370 b(to)2215 b Fq(F)154 b Fu(\()p Fq(x)p Fu(\))307 b(=)h Fq(F)38576 60693 y Fo(0)39318 60527 y Fu(+)246 b Fq(x)41058 60693 y Fo(1)41554 60527 y Fq(F)42266 60693 y Fo(1)43009 60527 y Fu(+)g Fq(:)184 b(:)g(:)248 b Fu(+)e Fq(x)47393 60693 y Fp(n)47995 60527 y Fq(F)48707 60693 y Fp(n)49617 60527 y Fr(\272)308 b Fu(0)34029 62188 y Fq(Ax)f Fu(=)g Fq(b;)26160 64712 y Fu(where)383 b Fq(x)332 b Fr(2)f Fk(R)32208 64310 y Fp(p)33104 64712 y Fu(and)384 b Fq(F)35983 64878 y Fp(i)36683 64712 y Fu(=)331 b Fq(F)38741 64310 y Fp(T)38587 65000 y(i)39770 64712 y Fr(2)h Fk(R)41639 64310 y Fp(p)p Fl(\243)p Fp(p)43299 64712 y Fu(.)536 b Fq(F)154 b Fu(\()p Fq(x)p Fu(\))332 b Fr(\272)f Fu(0)384 b(\(called)26160 66040 y(a)450 b(linear)g(matrix)h(inequalit)-31 b(y)-92 b(,)473 b(LMI\))449 b(restricts)g Fq(F)154 b Fu(\()p Fq(x)p Fu(\))449 b(to)h(b)31 b(e)26160 67368 y(con)-31 b(tained)393 b(in)f(the)f(p)31 b(ositiv)-31 b(e)393 b(semi-de\257nite)g(cone)e Fr(P)91 b Fu(.)559 b(LMI's)p 26160 68292 6401 45 v 27542 69041 a Fj(1)28003 69464 y Fi(S)408 b Fh(\265)352 b Fg(R)30905 69041 y Ff(d)31768 69464 y Fe(is)382 b(a)g(con)-28 b(v)g(ex)382 b(cone)g(if)g Fi(x;)172 b(y)387 b Fh(2)352 b Fi(S;)170 b(\270;)h(\271)352 b Fh(\270)g Fe(0)g Fh(\))g Fi(\270x)255 b Fe(+)26160 70571 y Fi(\271y)320 b Fh(2)284 b Fi(S)57 b Fe(.)p eop %%Page: 4 4 4 3 bop -1840 -322 a Fu(are)307 b(annotated)i(with)g Fr(\272)e Fu(instead)h(of)g Fr(\270)f Fu(to)g(distinguish)i(them)-1840 1006 y(from)352 b(normal)g(linear)g(inequalit)-31 b(y)354 b(statemen)-31 b(ts.)488 b(Notice)353 b(that)-1840 2335 y(the)514 b(ob)61 b(jectiv)-31 b(e)516 b(is)e(linear)g(in)g(the)g (unkno)-31 b(wns)515 b Fq(x)p Fu(,)550 b(and)514 b(that)-1840 3663 y(b)31 b(oth)334 b(the)g(LMI)e(and)i(the)g(equalit)-31 b(y)336 b(constrain)-31 b(t)335 b(are)e(linear)i(in)-1840 4992 y Fq(x)271 b Fu(\(e.g.,)293 b(with)273 b Fq(x)e Fu(b)31 b(eing)271 b(the)h(en)-31 b(tries)272 b(of)g Fq(F)424 b Fu(for)272 b(an)g(appropriate)-1840 6320 y(c)-31 b(hoice)370 b(of)g(the)f Fq(F)5329 6486 y Fp(i)5698 6320 y Fu('s\).)-1840 8312 y(W)-92 b(e)246 b(will)i(no)-31 b(w)248 b(sho)-31 b(w)248 b(ho)-31 b(w)247 b(optimizing)j(the)d (alignmen)-31 b(t)250 b(as)c(w)-31 b(ell)-1840 9641 y(as)293 b(optimizing)j(the)d(margin)h(can)f(b)31 b(e)292 b(cast)i(as)e(a)i (semi-de\257nite)-1840 10969 y(programming)469 b(problem.)784 b(F)-92 b(or)466 b(simplicit)-31 b(y)-92 b(,)493 b(w)-31 b(e)467 b(assume)g(in)-1840 12297 y(this)420 b(section)g(that)g(all)h (lab)31 b(els)420 b(are)f(kno)-31 b(wn.)644 b(Our)419 b(goal)i(is)e(to)-1840 13626 y(\257nd)485 b(the)h(optimal)j(em)-31 b(b)31 b(edding)486 b(\(i.e.,)517 b(the)486 b(optimal)j(k)-31 b(ernel)-1840 14954 y(matrix)361 b Fq(K)79 b Fu(\))360 b(suc)-31 b(h)360 b(that)g(a)g(measure)g(of)g(separation)h(b)31 b(et)-31 b(w)g(een)-1840 16283 y(those)370 b(t)-31 b(w)g(o)371 b(sets)d(of)i(data)g(is)f(maximized.)-1840 18275 y Fw(Alignmen)-35 b(t)599 b Fu(W)-92 b(e)599 b(w)-31 b(an)g(t)600 b(to)g(\257nd)f(the)g (k)-31 b(ernel)599 b(matrix)i Fq(K)-1840 19603 y Fu(whic)-31 b(h)370 b(is)f(maximally)k(aligned)e(with)f(the)g(set)f(of)h(lab)31 b(els)369 b Fq(y)40 b Fu(:)7322 21813 y(max)7953 22527 y Fp(K)11888 21533 y Fu(^)11596 21813 y Fq(A)p Fu(\()p Fq(S)r(;)184 b(K)18 b(;)184 b(y)40 b(y)16643 21356 y Fp(T)17342 21813 y Fu(\))4548 23849 y(sub)61 b(ject)370 b(to)2215 b Fq(K)387 b Fr(\272)307 b Fu(0)p Fq(:)-1840 26059 y Fu(This)387 b(is)g(equiv)-61 b(alen)-31 b(t)389 b(to)f(the)f(follo)-31 b(wing)391 b(optimization)g(prob-)-1840 27387 y(lem:)7730 29597 y(max)8361 30311 y Fp(K)12004 29597 y Fr(h)p Fq(K)18 b(;)184 b(y)40 b(y)15049 29139 y Fp(T)15747 29597 y Fr(i)4956 31694 y Fu(sub)61 b(ject)370 b(to)2215 b Fr(h)p Fq(K)18 b(;)184 b(K)79 b Fr(i)310 b Fu(=)d(1)12004 33355 y Fq(K)387 b Fr(\272)307 b Fu(0)p Fq(:)-1840 35564 y Fu(Giv)-31 b(en)438 b(the)h(linearit)-31 b(y)440 b(of)f(the)f(ob)61 b(jectiv)-31 b(e)440 b(in)f Fq(K)79 b Fu(,)456 b(w)-31 b(e)438 b(can)h(re-)-1840 36893 y(place)322 b(the)f(\257rst)f(constrain)-31 b(t)323 b(b)-31 b(y)321 b Fr(h)p Fq(K)18 b(;)184 b(K)79 b Fr(i)310 b(\267)d Fu(1)322 b(b)31 b(ecause)320 b(of)i(the)-1840 38221 y(maximization.)788 b(Using)467 b(the)g(Sc)-31 b(h)g(ur)466 b(complemen)-31 b(t)469 b(lemma)23662 37819 y Fo(2)-1840 39549 y Fu(for)610 b(this)h(constrain)-31 b(t,)673 b(one)610 b(can)h(write)g(the)g(optimization)-1840 40878 y(problem)370 b(as:)6659 43087 y(max)6825 43801 y Fp(A;K)10933 43087 y Fr(h)p Fq(K)18 b(;)184 b(y)40 b(y)13978 42630 y Fp(T)14675 43087 y Fr(i)3885 45335 y Fu(sub)61 b(ject)370 b(to)2215 b(trace\()p Fq(A)p Fu(\))309 b Fr(\267)e Fu(1)10933 46100 y Fn(\265)11842 46986 y Fq(A)1202 b(K)14893 46584 y Fp(T)11748 48315 y Fq(K)1758 b(I)15591 46100 y Fn(\266)16713 47661 y Fr(\272)307 b Fu(0)10933 49912 y Fq(K)387 b Fr(\272)307 b Fu(0)p Fq(:)-1840 52122 y Fu(Or,)369 b(b)-31 b(y)369 b(stac)-31 b(king)371 b(the)f(constrain)-31 b(ts)370 b(in)g(one)f(LMI:)1735 54331 y(max)1902 55045 y Fp(A;K)6009 54331 y Fr(h)p Fq(K)18 b(;)184 b(y)40 b(y)9054 53874 y Fp(T)9752 54331 y Fr(i)12564 b Fu(\(5\))-1038 58684 y(sub)61 b(ject)369 b(to)6009 55794 y Fn(0)6009 57743 y(B)6009 58407 y(B)6009 59115 y(@)7072 56681 y Fq(A)1202 b(K)10123 56279 y Fp(T)14490 56681 y Fq(O)3772 b(O)6978 58009 y(K)1758 b(I)4327 b(O)3772 b(O)7050 59337 y(O)1631 b(O)1559 b Fu(1)246 b Fr(\241)g Fu(trace)q(\()p Fq(A)p Fu(\))1180 b Fq(O)7050 60666 y(O)1631 b(O)4121 b(O)3700 b(K)20053 55794 y Fn(1)20053 57743 y(C)20053 58407 y(C)20053 59115 y(A)21329 58684 y Fr(\272)308 b Fu(0)p Fq(:)p -1840 61773 6401 45 v -458 62488 a Fj(2)3 62912 y Fe(Consider)341 b(the)h(partitioned)g(symmetric)h(matrix)5310 65542 y Fi(X)362 b Fe(=)285 b Fi(X)8523 65063 y Ff(T)9456 65542 y Fe(=)10536 64139 y Fd(\265)12196 64910 y Fi(A)1460 b(B)11844 66119 y(B)12669 65696 y Ff(T)14435 66119 y Fi(C)15801 64139 y Fd(\266)16726 65542 y Fi(:)-1840 68357 y(S)557 b Fe(=)500 b Fi(C)385 b Fh(\241)314 b Fi(B)3690 67934 y Ff(T)4338 68357 y Fi(A)5106 67934 y Fc(\241)p Fj(1)6207 68357 y Fi(B)520 b Fe(is)471 b(the)g(Sc)-28 b(h)g(ur)470 b(complemen)-28 b(t)472 b(of)f Fi(A)f Fe(in)h Fi(X)-1840 69464 y Fe(\(pro)-28 b(vided)512 b(det\()p Fi(A)p Fe(\))569 b Fh(6)p Fe(=)g(0\).)966 b(The)512 b(Sc)-28 b(h)g(ur)510 b(complemen)-28 b(t)513 b(lemma)-1840 70571 y(then)342 b(sa)-28 b(ys:)455 b(if)342 b Fi(A)284 b Fh(\302)h Fe(0,)341 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b(complemen)-31 b(t)330 b(lemma\),)26160 67329 y(suc)-31 b(h)369 b(that)30378 68512 y Fn(\265)33683 69396 y Fq(G)p Fu(\()p Fq(K)79 b Fu(\))3046 b Fq(e)246 b Fu(+)g Fq(\272)317 b Fu(+)246 b Fq(\270y)31746 70726 y Fu(\()p Fq(e)h Fu(+)f Fq(\272)316 b Fu(+)246 b Fq(\270y)40 b Fu(\))37674 70324 y Fp(T)41811 70726 y Fq(t)45098 68512 y Fn(\266)46221 70072 y Fr(\272)307 b Fu(0)p eop %%Page: 5 5 5 4 bop -1840 -322 a Fu(holds.)614 b(Stac)-31 b(king)412 b(all)f(constrain)-31 b(ts)411 b(in)f(one)g(single)g(LMI,)g(\(7\))-1840 1006 y(can)369 b(b)31 b(e)369 b(expressed)f(as)h(a)h(standard)f(SDP)g (\(4\):)898 3996 y(min)374 4718 y Fp(K)7 b(;t;\270;\272)3821 3996 y Fq(t)18552 b Fu(\(8\))374 6253 y(s.t.)370 b(trace)q(\()p Fq(K)79 b Fu(\))308 b(=)g Fq(c;)374 7017 y Fn(0)374 8965 y(B)374 9629 y(B)374 10338 y(@)1343 7902 y Fq(K)4061 b(O)6109 b(O)4550 b(O)1415 9230 y(O)3147 b(G)p Fu(\()p Fq(K)79 b Fu(\))3046 b Fq(e)246 b Fu(+)g Fq(\272)317 b Fu(+)245 b Fq(\270y)2463 b(O)1415 10560 y(O)1210 b Fu(\()p 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b(in)h(\(3\),)412 b(b)31 b(ecause)402 b Fq(K)481 b Fu(is)402 b(no)-31 b(w)403 b(indeed)g(con-)26160 56623 y(sidered)497 b(as)h(a)g(linear)h(com)-31 b(bination)501 b Fq(K)601 b Fu(=)45151 55793 y Fn(P)46319 56067 y Fp(m)46319 56956 y(i)p Fo(=1)47993 56623 y Fq(\271)48660 56789 y Fp(i)49028 56623 y Fq(K)49968 56789 y Fp(i)50835 56623 y Fu(for)26160 57952 y(a)504 b(\257xed)g(set)g Fr(f)p Fq(K)33383 58118 y Fo(1)33880 57952 y Fq(;)184 b(:)g(:)g(:)k(;)c(K) 37279 58118 y Fp(m)38121 57952 y Fr(g)p Fu(.)897 b(The)504 b(margin)h(is)f(optimized)26160 59280 y(o)-31 b(v)g(er)317 b(the)g(lab)31 b(elled)318 b(data)g(\(use)e(of)h Fq(K)41276 59446 y Fp(i;tr)42677 59280 y Fu(\),)328 b(while)317 b(the)g(p)31 b(ositiv)-31 b(e)26160 60609 y(semi-de\257niteness)320 b(and)f(the)h(trace)f(constrain)-31 b(t)321 b(are)e(imp)31 b(osed)26160 61937 y(for)369 b(the)h(en)-31 b(tire)369 b(k)-31 b(ernel)370 b(matrix)h Fq(K)448 b Fu(\(use)369 b(of)h Fq(K)45665 62103 y Fp(i)46033 61937 y Fu(\).)26160 63929 y(This)295 b(giv)-31 b(es)295 b(a)g(new)g(transduction)g(metho)31 b(d)296 b(for)e(hard)h(margin)26160 65258 y(SVMs)353 b(\(real)h(transduction)h(with)g(rank)f(one)g(matrices)g(from)26160 66586 y(eigen)-31 b(v)g(ectors)379 b(in)e(\(10\);)383 b(otherwise)378 b(not)g(fully)g(transduction,)26160 67914 y(but)385 b(this)g(do)31 b(esn't)384 b(seem)h(a)f(disadv)-61 b(an)-31 b(tage\).)542 b(V)-92 b(apnik's)385 b(orig-)26160 69243 y(inal)427 b(straigh)-31 b(tforw)g(ard)429 b(transduction)f (approac)-31 b(h)427 b(scales)f(ex-)26160 70571 y(p)31 b(onen)-31 b(tially)255 b(in)e(the)g(n)-31 b(um)g(b)31 b(er)252 b(of)h(test)g(samples.)454 b(This)253 b(o\256ers)f(a)p eop %%Page: 6 6 6 5 bop -1840 -322 a Fu(more)318 b(practice)f(transduction)i(metho)31 b(d)318 b(that)h(is)e(p)31 b(olynomial)-1840 1006 y(time)409 b(-)e(sp)31 b(ecial)409 b(purp)31 b(ose)406 b(metho)31 b(ds)409 b(exploiting)h(the)e(partic-)-1840 2335 y(ular)367 b(structure)g(of)g(the)h(problem)f(should)h(result)f(in)g(v)-31 b(ery)367 b(ef-)-1840 3663 y(\257cien)-31 b(t)389 b(algorithms.)552 b(Before)388 b(giving)i(empirical)g(results,)i(w)-31 b(e)-1840 4992 y(pro)g(vide)370 b(a)f(fuller)h(in)-31 b(v)g(estigation)373 b(of)d(the)f(error)g(b)31 b(ound)369 b(\(3\).)-1840 8165 y Ft(5.)498 b(Error)i(Bounds)g(for)e(T)-125 b(ransduction)-1840 10318 y Fu(In)371 b(the)h(problem)h(of)f (transduction,)i(w)-31 b(e)372 b(ha)-31 b(v)g(e)373 b(access)e(to)i (the)-1840 11646 y(unlab)31 b(elled)482 b(test)f(data,)510 b(as)480 b(w)-31 b(ell)482 b(as)e(the)h(lab)31 b(elled)482 b(training)-1840 12975 y(data,)304 b(and)286 b(the)h(aim)g(is)e(to)i (optimize)h(accuracy)e(in)h(predicting)-1840 14303 y(the)349 b(test)g(data.)487 b(W)-92 b(e)347 b(assume)i(that)h(the)e(data)i(are)e (\257xed,)354 b(and)-1840 15631 y(that)396 b(the)e(order)g(is)g(c)-31 b(hosen)395 b(randomly)-92 b(,)402 b(yielding)396 b(a)f(random)-1840 16960 y(partition)379 b(in)-31 b(to)378 b(training)g(and)f(test)g (sets.)514 b(F)-92 b(or)376 b(con)-31 b(v)g(enience,)-1840 18288 y(w)g(e)443 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y Fu(1)p 26293 20038 V 26293 21052 a Fq(n)27801 18909 y Fp(n)27275 19241 y Fn(X)27357 21598 y Fp(i)p Fo(=1)29058 20292 y Fq(\301)p Fu(\()p Fq(Y)30791 20458 y Fp(i)31159 20292 y Fq(f)119 b Fu(\()p Fq(X)33167 20458 y Fp(i)33537 20292 y Fu(\)\)+)35908 19544 y(1)p 35391 20038 1587 45 v 35391 20259 a Fr(p)p 36314 20259 665 45 v 36314 21057 a Fq(n)37296 18400 y Fn(\303)38172 20292 y Fu(4)246 b(+)g(2)40631 19292 y Fn(p)p 41739 19292 3932 45 v 41739 20292 a Fu(log)17 b(\(1)p Fq(=\261)42 b Fu(\))247 b(+)47023 18270 y Fn(s)p 48130 18270 4548 45 v 48263 19544 a Fu(trace)q(\()p Fq(K)79 b Fu(\))p 48263 20038 4282 45 v 49506 21052 a Fq(n\260)50804 20732 y Fo(2)52678 18400 y Fn(!)53739 20292 y Fq(:)26160 23400 y Fs(Mor)-57 b(e)g(over,)458 b(for)446 b(any)g(set)f Fr(f)p Fq(K)38686 23566 y Fo(1)39183 23400 y Fq(;)184 b(:)g(:)g(:)k(;)c(K)42582 23566 y Fp(m)43424 23400 y Fr(g)446 b Fs(of)g(kernel)g(matri-)26160 24728 y(c)-57 b(es,)375 b(de\257ne)369 b Fq(C)387 b Fu(=)307 b Fw(E)184 b Fu(sup)q(\()p 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y(comp)31 b(onen)-31 b(ts)511 b Fq(K)5335 66088 y Fp(i)5703 65922 y Fu(.)913 b(This)510 b(supp)31 b(orts)509 b(the)h(use)e(of)i(the)g(error)-1840 67250 y(b)31 b(ound)369 b(\(3\).)-1840 69243 y(The)595 b(result)f(for)g(the)h (heart)f(dataset)i(is)e(w)-31 b(orth)595 b(noting|)-1840 70571 y(although)314 b Fq(K)3655 70737 y Fo(1)4464 70571 y Fu(and)e Fq(K)7499 70737 y Fo(3)8307 70571 y Fu(fail)i(to)e(giv)-31 b(e)313 b(rise)f(to)g(a)h(linearly)g(sepa-)26160 -322 y(rable)264 b(em)-31 b(b)31 b(edding,)286 b(com)-31 b(bining)266 b(them)e(linearly)h(with)g Fq(K)49801 -156 y Fo(2)50561 -322 y Fu(im-)26160 1006 y(pro)-31 b(v)g(es)369 b(margin)i(and)e(test)h (set)f(p)31 b(erformance)369 b(signi\257can)-31 b(tly)-92 b(.)26160 4180 y Ft(7.)498 b(Conclusions)26160 6333 y Fu(In)553 b(this)g(pap)31 b(er)553 b(w)-31 b(e)554 b(ha)-31 b(v)g(e)554 b(presen)-31 b(ted)553 b(a)g(new)h(metho)31 b(d)554 b(for)26160 7661 y(learning)522 b(a)g(k)-31 b(ernel)521 b(matrix)i(from)e(data,)561 b(that)522 b(mak)-31 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(mac)-31 b(hine)274 b(learning)f(concerns)f(merge)26160 20945 y(to)436 b(pro)-31 b(vide)435 b(a)g(p)31 b(o)-31 b(w)g(erful)436 b(metho)31 b(d)436 b(for)f(learning)h(the)f(k)-31 b(ernel)26160 22273 y(matrix)371 b(with)f(SDP)-92 b(.)26160 24266 y(The)516 b(learning)i(pro)31 b(cess)515 b(is)h(conducted)h(in)f (a)h(transductiv)-31 b(e)26160 25594 y(setting|using)387 b(the)e(lab)31 b(elled)386 b(data)g(one)f(can)g(learn)g(a)g(go)31 b(o)g(d)26160 26922 y(em)-31 b(b)31 b(edding,)564 b(whic)-31 b(h)525 b(can)g(then)f(b)31 b(e)524 b(applied)h(to)g(the)f(unla-)26160 28251 y(b)31 b(elled)376 b(part)f(of)h(the)f(data.)512 b(After)376 b(deriving)g(a)g(new)f(general-)26160 29579 y(ization)299 b(b)31 b(ound)297 b(that)h(giv)-31 b(es)298 b(rise)e(to)i(a)f(con)-31 b(v)g(ex)299 b(cost)e(function,)26160 30907 y(w)-31 b(e)408 b(imp)31 b(ose)408 b(con)-31 b(v)g(ex)409 b(constrain)-31 b(ts)409 b(that)g(con)-31 b(trol)409 b(the)f(capac-)26160 32236 y(it)-31 b(y)512 b(of)f(the)g(searc)-31 b(h)510 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Fq(g)40 b Fu(\()p Fq(X)43115 46718 y Fp(i)43483 46552 y Fq(;)184 b(Y)44617 46718 y Fp(i)44987 46552 y Fu(\))p Fq(;)30283 49957 y Fu(^)30142 50240 y Fw(E)30978 50406 y Fo(2)31475 50240 y Fq(g)40 b Fu(\()p Fq(X)25 b(;)184 b(Y)247 b Fu(\))1108 b(=)38490 49491 y(1)p 38434 49985 V 38434 50999 a Fq(n)39942 48856 y Fp(n)39416 49188 y Fn(X)39499 51546 y Fp(i)p Fo(=1)41200 50240 y Fq(g)40 b Fu(\()p Fq(X)43115 50406 y Fp(n)p Fo(+)p Fp(i)44710 50240 y Fq(;)184 b(Y)45844 50406 y Fp(n)p Fo(+)p Fp(i)47440 50240 y Fu(\))p Fq(:)26160 54058 y Fu(The)434 b(pro)31 b(of)434 b(of)g(the)g(\257rst)f(part)h(in)-31 b(v)g(olv)g(es)435 b(the)f(follo)-31 b(wing)438 b(\257v)-31 b(e)26160 55387 y(steps:)26160 57379 y Fw(Step)475 b(1.)621 b Fu(F)-92 b(or)412 b(an)-31 b(y)413 b(class)f Fq(F)566 b Fu(of)413 b(real)f(functions)i(de\257ned)e(on)26160 58708 y Fr(X)162 b Fu(,)29031 61558 y(sup)28966 62496 y Fp(f)87 b Fl(2)p Fp(F)30946 61558 y Fu(er\()p Fq(f)119 b Fu(\))247 b Fr(\241)34888 61275 y Fu(^)34747 61558 y Fw(E)35583 61724 y Fo(1)36080 61558 y Fq(\301)36740 61724 y Fp(\260)37307 61558 y Fu(\()p Fq(Y)f(f)119 b Fu(\()p Fq(X)87 b Fu(\)\))30073 64252 y Fr(\267)1172 b Fu(sup)32041 65190 y Fp(f)87 b Fl(2)p Fp(F)34163 63969 y Fu(^)34021 64252 y Fw(E)34857 64418 y Fo(2)35354 64252 y Fq(\301)36014 64418 y Fp(\260)36582 64252 y Fu(\()p Fq(Y)246 b(f)119 b Fu(\()p Fq(X)87 b Fu(\)\))247 b Fr(\241)42352 63969 y Fu(^)42210 64252 y Fw(E)43046 64418 y Fo(1)43543 64252 y Fq(\301)44203 64418 y Fp(\260)44771 64252 y Fu(\()p Fq(Y)f(f)119 b Fu(\()p Fq(X)87 b Fu(\)\))p Fq(:)26160 67914 y Fu(T)-92 b(o)442 b(see)f(this,)461 b(notice)443 b(that)g(er\()p Fq(f)119 b Fu(\))442 b(is)f(the)h(a)-31 b(v)g(erage)443 b(o)-31 b(v)g(er)443 b(the)26160 69243 y(test)436 b(set)f(of)i(the)e (indicator)j(function)f(of)f Fq(Y)245 b(f)119 b Fu(\()p Fq(X)87 b Fu(\))420 b Fr(\267)e Fu(0,)453 b(and)26160 70571 y(that)370 b Fq(\301)29218 70737 y Fp(\260)29786 70571 y Fu(\()p Fq(Y)246 b(f)119 b Fu(\()p Fq(X)87 b Fu(\)\))371 b(b)31 b(ounds)368 b(this)i(function.)p eop %%Page: 8 8 8 7 bop -1840 -322 a Fw(Step)425 b(2.)492 b Fu(F)-92 b(or)369 b(an)-31 b(y)370 b(class)f Fq(G)g Fu(of)h([0)p Fq(;)184 b Fu(1]-v)-61 b(alued)373 b(functions,)-386 2391 y(Pr)985 831 y Fn(\265)1839 2391 y Fu(sup)1800 3321 y Fp(g)28 b Fl(2)p Fp(G)3871 2109 y Fu(^)3730 2391 y Fw(E)4566 2557 y Fo(2)5063 2391 y Fq(g)285 b Fr(\241)7125 2109 y Fu(^)6983 2391 y Fw(E)7819 2557 y Fo(1)8316 2391 y Fq(g)348 b Fr(\270)307 b Fw(E)11380 831 y Fn(\265)12235 2391 y Fu(sup)12196 3321 y Fp(g)28 b Fl(2)p Fp(G)14267 2109 y Fu(^)14126 2391 y Fw(E)14962 2557 y Fo(2)15458 2391 y Fq(g)286 b Fr(\241)17521 2109 y Fu(^)17379 2391 y Fw(E)18215 2557 y Fo(1)18712 2391 y Fq(g)19280 831 y Fn(\266)20341 2391 y Fu(+)246 b Fq(\262)21897 831 y Fn(\266)721 5675 y Fr(\267)1107 b Fu(exp)4564 4114 y Fn(\265)5512 4926 y Fr(\241)p Fq(\262)6822 4524 y Fo(2)7319 4926 y Fq(n)p 5512 5420 2472 45 v 6471 6434 a Fu(4)8116 4114 y Fn(\266)9115 5675 y Fq(;)-1840 8327 y Fu(where)518 b(the)h(exp)31 b(ectation)520 b(is)e(o)-31 b(v)g(er)519 b(the)g(random)g(p)31 b(erm)-31 b(uta-)-1840 9655 y(tion.)670 b(This)428 b(follo)-31 b(ws)430 b(from)f(McDiarmid's)f(inequalit)-31 b(y)431 b(\(Mc-)-1840 10983 y(Diarmid,)556 b(1989\).)941 b(T)-92 b(o)518 b(see)f(this,)556 b(w)-31 b(e)518 b(need)g(to)g (de\257ne)g(the)-1840 12312 y(random)403 b(p)31 b(erm)-31 b(utation)404 b Fq(\274)441 b Fu(using)403 b(a)f(set)g(of)h(2)p Fq(n)f Fu(indep)31 b(enden)-31 b(t)-1840 13640 y(random)607 b(v)-61 b(ariables.)1205 b(T)-92 b(o)607 b(this)f(end,)666 b(c)-31 b(ho)31 b(ose)607 b Fq(\274)19529 13806 y Fo(1)20025 13640 y Fq(;)184 b(:)g(:)g(:)k(;)c(\274)23115 13806 y Fo(2)p Fp(n)-1840 14968 y Fu(uniformly)501 b(at)e(random)h(from)f(the)g (in)-31 b(terv)-61 b(al)501 b([0)p Fq(;)184 b Fu(1].)884 b(These)-1840 16297 y(are)472 b(almost)h(surely)f(distinct.)802 b(F)-92 b(or)471 b Fq(j)542 b Fu(=)478 b(1)p Fq(;)184 b(:)g(:)g(:)k(;)c Fu(2)p Fq(n)p Fu(,)500 b(de\257ne)-1840 17625 y Fq(\274)40 b Fu(\()p Fq(j)63 b Fu(\))387 b(=)g Fr(jf)p Fq(i)h Fu(:)f Fq(\274)4799 17791 y Fp(i)5553 17625 y Fr(\267)g Fq(\274)7432 17791 y Fp(j)7898 17625 y Fr(gj)p Fu(,)430 b(that)418 b(is,)429 b Fq(\274)40 b Fu(\()p Fq(j)63 b Fu(\))417 b(is)f(the)h(p)31 b(osition)418 b(of)-1840 18953 y Fq(\274)-1209 19119 y Fp(j)-343 18953 y Fu(when)400 b(the)g(random)h(v)-61 b(ariables)400 b(are)g(ordered)f (b)-31 b(y)400 b(size.)585 b(It)-1840 20432 y(is)436 b(easy)f(to)i(see)e(that,)454 b(for)435 b(an)-31 b(y)437 b Fq(g)40 b Fu(,)13293 20150 y(^)13151 20432 y Fw(E)13987 20598 y Fo(2)14484 20432 y Fq(g)330 b Fr(\241)16635 20150 y Fu(^)16493 20432 y Fw(E)17329 20598 y Fo(1)17826 20432 y Fq(g)475 b Fu(c)-31 b(hanges)437 b(b)-31 b(y)-1840 21761 y(no)365 b(more)g(than)g(2)p Fq(=n)h Fu(when)f(one)g(of)g(the)g Fq(\274)15873 21927 y Fp(i)16607 21761 y Fu(c)-31 b(hanges.)491 b(McDi-)-1840 23089 y(armid's)370 b(inequalit)-31 b(y)372 b(implies)f(the)e(result.)-1840 25082 y Fw(Step)425 b(3.)492 b Fu(F)-92 b(or)369 b(an)-31 b(y)370 b(class)f Fq(G)g Fu(of)h([0)p Fq(;)184 b Fu(1]-v)-61 b(alued)373 b(functions,)1990 27795 y Fw(E)3010 26234 y Fn(\265)3866 27795 y Fu(sup)3826 28724 y Fp(g)28 b Fl(2)p Fp(G)5897 27512 y Fu(^)5756 27795 y Fw(E)6592 27961 y Fo(2)7089 27795 y Fq(g)286 b Fr(\241)9151 27512 y Fu(^)9010 27795 y Fw(E)9846 27961 y Fo(1)10343 27795 y Fq(g)10911 26234 y Fn(\266)12033 27795 y Fr(\267)13441 27515 y Fu(^)13201 27795 y Fq(R)14042 27961 y Fo(2)p Fp(n)15085 27795 y Fu(\()p Fq(G)p Fu(\))248 b(+)18819 27046 y(4)p 18302 27540 1587 45 v 18302 27762 a Fr(p)p 19225 27762 665 45 v 19225 28559 a Fq(n)20022 27795 y(;)-1840 30873 y Fu(where)1777 30593 y(^)1537 30873 y Fq(R)2378 31039 y Fo(2)p Fp(n)3421 30873 y Fu(\()p Fq(G)p Fu(\))602 b(=)e Fw(E)184 b Fu(sup)9902 31144 y Fp(g)28 b Fl(2)p Fp(G)12073 30437 y Fo(1)p 12020 30618 548 45 v 12020 31255 a Fp(n)12884 30043 y Fn(P)14053 30316 y Fo(2)p Fp(n)14053 31205 y(i)p Fo(=1)15727 30873 y Fq(\276)16360 31039 y Fp(i)16728 30873 y Fq(g)40 b Fu(\()p Fq(X)18643 31039 y Fp(i)19012 30873 y Fu(\))p Fq(;)545 b Fu(and)h(the)-1840 32201 y(exp)31 b(ectation)481 b(is)e(o)-31 b(v)g(er)479 b(the)g(indep)31 b(enden)-31 b(t,)507 b(uniform,)h Fr(f\247)p Fu(1)p Fr(g)p Fu(-)-1840 33530 y(v)-61 b(alued)551 b(random)h(v)-61 b(ariables)551 b Fq(\276)11465 33696 y Fo(1)11961 33530 y Fq(;)184 b(:)g(:)g(:)k(;)c (\276)15053 33696 y Fo(2)p Fp(n)16097 33530 y Fu(.)1037 b(This)551 b(result)g(is)-1840 34858 y(essen)-31 b(tially)269 b(Lemma)g(3)f(of)g(Bartlett)h(and)f(Mendelson)g(\(2001\);)-1840 36186 y(that)526 b(lemma)f(con)-31 b(tained)527 b(a)d(similar)i(b)31 b(ound)524 b(for)g(i.i.d.)j Fq(X)23484 36352 y Fp(i)23853 36186 y Fu(,)-1840 37515 y(but)433 b(the)g(same)g(argumen)-31 b(t)434 b(holds)f(for)g(\257xed)g Fq(X)18451 37681 y Fp(i)18819 37515 y Fu(,)449 b(randomly)-1840 38843 y(p)31 b(erm)-31 b(uted.)-1840 40836 y Fw(Step)592 b(4.)928 b Fu(If)514 b(the)g(class)h Fq(F)668 b Fu(of)514 b(real-v)-61 b(alued)516 b(functions)f(de-)-1840 42164 y(\257ned)431 b(on)g 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48600 y Fp(K)11793 48434 y Fu(of)g(k)-31 b(ernel)469 b(expansions)g(\(13\))-1840 49763 y(satisfying)371 b(\(14\),)5194 52053 y(^)4954 52333 y Fq(R)5795 52499 y Fo(2)p Fp(n)6838 52333 y Fu(\()p Fq(F)7980 52499 y Fp(K)8834 52333 y Fu(\))308 b Fr(\267)10928 51584 y Fu(1)p 10873 52078 665 45 v 10873 53092 a Fq(n)11670 51332 y Fn(p)p 12777 51332 4282 45 v 1001 x Fu(trace)q(\()p Fq(K)79 b Fu(\))q Fq(:)-1840 54754 y Fu(This)370 b(is)f(Lemma)h(26)g(of)g(Bartlett)h(and)e (Mendelson)g(\(2001\).)-1840 56747 y(Com)-31 b(bining)562 b(giv)-31 b(es)561 b(the)e(\257rst)g(part)h(of)g(the)f(theorem.)1064 b(T)-92 b(o)-1840 58075 y(pro)-31 b(v)g(e)672 b(the)f(second)g(part,) 748 b(notice)672 b(\(as)g(in)f(the)h(pro)31 b(of)671 b(of)-1840 59403 y(Lemma)370 b(26)g(of)g(Bartlett)h(and)e(Mendelson,)h (2001\))i(that)548 62225 y(^)308 62505 y Fq(R)1149 62671 y Fo(2)p Fp(n)2193 62505 y Fu(\()p Fq(F)3335 62671 y Fp(B)4097 62505 y Fu(\))1107 b(=)7791 61756 y(1)p 7735 62251 665 45 v 7735 63265 a Fq(n)8532 62505 y Fw(E)184 b Fu(sup)10116 63443 y Fp(\257)11922 62505 y Fu(sup)11404 63486 y Fl(k)p Fp(w)21 b Fl(k\267)p Fo(1)14106 62505 y Fr(h)p Fq(w)30 b(;)16156 61122 y Fo(2)p Fp(n)15851 61454 y Fn(X)15933 63811 y Fp(i)p Fo(=1)17634 62505 y Fq(\276)18267 62671 y Fp(i)18635 62505 y Fu(\251\()p Fq(X)20781 62671 y Fp(i)21151 62505 y Fu(\))p Fr(i)5634 66359 y Fu(=)7791 65610 y(1)p 7735 66104 V 7735 67118 a Fq(n)8532 66359 y Fw(E)184 b Fu(sup)10116 67296 y Fp(\257)11404 64421 y Fn(\260)11404 65086 y(\260)11404 65750 y(\260)11404 66414 y(\260)11404 67078 y(\260)12325 64975 y Fo(2)p Fp(n)12019 65307 y Fn(X)12102 67665 y Fp(i)p Fo(=1)13803 66359 y Fq(\276)14436 66525 y Fp(i)14804 66359 y Fu(\251\()p Fq(X)16950 66525 y Fp(i)17319 66359 y Fu(\))17749 64421 y Fn(\260)17749 65086 y(\260)17749 65750 y(\260)17749 66414 y(\260)17749 67078 y(\260)5634 69680 y Fr(\267)7791 68931 y Fu(1)p 7735 69426 V 7735 70440 a Fq(n)8532 68512 y Fn(r)p 9639 68512 5547 45 v 1168 x Fw(E)g Fu(sup)11223 70618 y Fp(\257)12511 69680 y Fq(\276)13184 69361 y Fl(0)13494 69680 y Fq(K)79 b(\276)40 b(;)26160 -322 y Fu(where)433 b Fq(\276)454 b Fu(=)414 b(\()p Fq(\276)32850 -156 y Fo(1)33347 -322 y Fq(;)184 b(:)g(:)g(:)k(;)c(\276)36439 -156 y Fo(2)p Fp(n)37483 -322 y Fu(\))433 b(is)h(the)f(v)-31 b(ector)434 b(of)g(Rademac)-31 b(her)26160 1006 y(random)623 b(v)-61 b(ariables,)686 b(and)622 b(the)g(suprem)-31 b(um)622 b(is)g(o)-31 b(v)g(er)623 b Fq(\257)787 b Fu(=)26160 2335 y(\()p Fq(\257)27216 2501 y Fo(1)27713 2335 y Fq(;)184 b(:)g(:)g(:)k(;)c(\257)30798 2501 y Fp(m)31640 2335 y Fu(\))540 b(for)f(whic)-31 b(h)540 b(the)g(matrix)h Fq(K)670 b Fu(=)46721 1505 y Fn(P)47889 1778 y Fp(m)47889 2667 y(j)45 b Fo(=1)49661 2335 y Fq(\257)50287 2501 y Fp(j)50753 2335 y Fq(K)51693 2501 y Fp(j)26160 3663 y Fu(satis\257es)557 b(the)g(conditions)i Fq(K)700 b Fr(\272)620 b Fu(0)558 b(and)f(trace\()p Fq(K)79 b Fu(\))622 b Fr(\267)e Fq(B)56 b Fu(.)26160 4992 y(No)-31 b(w,)321 b(trace\()p Fq(K)79 b Fu(\))309 b(=)34697 4161 y Fn(P)35865 4435 y Fp(m)35865 5324 y(j)45 b Fo(=1)37637 4992 y Fq(\257)38263 5158 y Fp(j)38729 4992 y Fu(trace)q(\()p Fq(K)42501 5158 y Fp(j)42968 4992 y Fu(\))p Fq(;)307 b Fu(and)g(eac)-31 b(h)307 b(trace)g(in)26160 6320 y(the)283 b(sum)g(is)f(p)31 b(ositiv)-31 b(e,)302 b(so)283 b(the)g(suprem)-31 b(um)283 b(m)-31 b(ust)283 b(b)31 b(e)282 b(ac)-31 b(hiev)g(ed)26160 7648 y(for)316 b(trace)q(\()p Fq(K)79 b Fu(\))308 b(=)g Fq(B)56 b Fu(.)474 b(So)316 b(w)-31 b(e)317 b(can)f(write)h(that)45637 7368 y(^)45397 7648 y Fq(R)46238 7814 y Fo(2)p Fp(n)47281 7648 y Fu(\()p Fq(F)48423 7814 y Fp(B)49185 7648 y Fu(\))g(is)f(no) 26160 8977 y(more)370 b(than)35324 9176 y Fr(p)p 36247 9176 6442 45 v 36247 10007 a Fq(B)56 b Fw(E)184 b Fu(sup)g Fq(\276)40687 9687 y Fl(0)40997 10007 y Fq(K)79 b(\276)p 35324 10501 7364 45 v 38674 11515 a(n)42821 10756 y(;)26160 13120 y Fu(where)398 b(the)h(suprem)-31 b(um)398 b(is)h(o)-31 b(v)g(er)399 b Fq(K)435 b Fu(=)42880 12290 y Fn(P)44049 13452 y Fp(j)44700 13120 y Fq(\257)45326 13286 y Fp(j)45792 13120 y Fq(K)46732 13286 y Fp(j)47597 13120 y Fu(satisfying)26160 14449 y Fq(K)492 b Fr(\272)414 b Fu(0)432 b(and)h(trace)q(\()p Fq(K)79 b Fu(\))414 b(=)f(1.)683 b(De\257ning)433 b Fq(C)493 b Fu(=)413 b Fw(E)184 b Fu(sup)h Fq(\276)50159 14047 y Fl(0)50468 14449 y Fq(K)79 b(\276)26160 15777 y Fu(giv)-31 b(es)370 b(the)g(result.)26160 17770 y(Notice)667 b(that)g Fq(\276)33299 17368 y Fl(0)33609 17770 y Fq(K)79 b(\276)705 b Fu(is)666 b(no)g(more)g(than)g Fq(\270)p Fr(k)p Fq(\276)40 b Fr(k)47581 17368 y Fo(2)48880 17770 y Fu(=)801 b Fq(n\270)p Fu(,)26160 19098 y(where)618 b Fq(\270)f Fu(is)h(the)g(maxim)-31 b(um)621 b(eigen)-31 b(v)-61 b(alue)620 b(of)e Fq(K)79 b Fu(.)1239 b(Using)26160 20426 y Fq(\270)307 b Fr(\267)h Fu(trace\()p Fq(K)79 b Fu(\))309 b(=)e(1)370 b(sho)-31 b(ws)369 b(that)i Fq(C)387 b Fr(\267)307 b Fq(n)p Fu(.)26160 23600 y Ft(Ac)-42 b(kno)g(wledgemen)g(ts)26160 25753 y Fu(W)-92 b(e)639 b(w)-31 b(ould)642 b(lik)-31 b(e)641 b(to)g(ac)-31 b(kno)g(wledge)643 b(supp)31 b(ort)639 b(from)i(ONR)26160 27081 y(MURI)369 b(N00014-00-1-0637)374 b(and)c(NSF)f(gran)-31 b(t)370 b(I)31 b(IS-9988642.)26160 30255 y Ft(References)26160 32407 y Fu(Bartlett,)454 b(P)-92 b(.)436 b(L.)f(&)h(Mendelson,)452 b(S.)436 b(\(2001\).)693 b Fs(R)-57 b(ademacher)27267 33736 y(and)472 b(gaussian)g(c)-57 b(omplexities:)661 b(R)-28 b(isk)473 b(b)-57 b(ounds)472 b(and)g(struc-)27267 35064 y(tur)-57 b(al)352 b(r)-57 b(esults.)476 b Fu(T)-92 b(ec)-31 b(hnical)323 b(Rep)31 b(ort,)331 b(Australian)323 b(National)27267 36393 y(Univ)-31 b(ersit)g(y)-92 b(.)26160 38385 y(Bennett,)569 b(K.)528 b(P)-92 b(.)528 b(&)g(Bredensteiner,)567 b(E.)528 b(J.)g(\(2000\).)971 b(Du-)27267 39713 y(alit)-31 b(y)426 b(and)f(geometry)h(in)e(SVM)g (classi\257ers.)658 b Fs(Pr)-57 b(o)g(c)g(e)g(e)g(dings)27267 41042 y(of)348 b(the)g(Sevente)-57 b(enth)349 b(International)f(Confer) -57 b(enc)g(e)351 b(on)d(Ma-)27267 42370 y(chine)396 b(L)-57 b(e)g(arning)p Fu(.)494 b(San)369 b(F)-92 b(rancisco:)491 b(Morgan)369 b(Kaufmann.)26160 44363 y(Cristianini,)494 b(N.)467 b(&)f(Sha)-31 b(w)g(e-T)-92 b(a)-31 b(ylor)469 b(J.)e(\(2000\).)787 b Fs(A)-28 b(n)485 b(Intr)-57 b(o-)27267 45691 y(duction)547 b(to)f(Supp)-57 b(ort)547 b(V)-85 b(e)-57 b(ctor)547 b(Machines)p Fu(.)983 b(Cam)-31 b(bridge:)27267 47019 y(Cam)g(bridge)371 b(Univ)-31 b(ersit)g(y)371 b(Press.)26160 49012 y(Cristianini,)518 b(N.,)e(Sha)-31 b(w)g(e-T)-92 b(a)-31 b(ylor)489 b(J.,)516 b(Kandola)488 b(J.)e(&)g(Elis-)27267 50340 y(see\256)497 b(A.)h(\(2001\).)881 b(On)497 b(k)-31 b(ernel)498 b(target)h(alignmen)-31 b(t.)881 b Fs(A)-57 b(d-)27267 51669 y(vanc)g(es)531 b(in)g(Neur)-57 b(al)531 b(Information)g(Pr)-57 b(o)g(c)g(essing)531 b(Systems,)27267 52997 y(14)p Fu(.)492 b(Cam)-31 b(bridge,)372 b(MA:)d(MIT)g(Press.) 26160 54989 y(Ledoux,)534 b(M.)501 b(&)f(T)-92 b(alagrand,)536 b(M.)500 b(\(1991\).)890 b Fs(Pr)-57 b(ob)g(ability)518 b(in)27267 56318 y(Banach)j(Sp)-57 b(ac)g(es:)759 b(Isop)-57 b(erimetry)520 b(and)h(Pr)-57 b(o)g(c)g(esses.)898 b Fu(NY:)27267 57646 y(Springer-V)-92 b(erlag.)26160 59639 y(McDiarmid,)291 b(C.)270 b(\(1989\).)463 b(On)269 b(the)h(metho)31 b(d)271 b(of)g(b)31 b(ounded)269 b(dif-)27267 60967 y(ferences.)674 b(In)429 b Fs(Surveys)452 b(in)h(Combinatorics)f(1989)p Fu(,)444 b(pages)27267 62295 y(148{188.)496 b(Cam)-31 b(bridge:)494 b(Cam)-31 b(bridge)372 b(Univ)-31 b(ersit)g(y)370 b(Press.)26160 64288 y(Sc)-31 b(h\304)-553 b(olk)-31 b(opf,)345 b(B.)336 b(&)f(Smola,)344 b(A.)336 b(\(2002\).)484 b Fs(L)-57 b(e)g(arning)367 b(with)e(Ker-)27267 65616 y(nels.)493 b Fu(Cam)-31 b(bridge,)371 b(MA:)f(MIT)e(Press.)26160 67609 y(V)-92 b(anden)-31 b(b)31 b(erghe,)319 b(L.)306 b(&)g(Bo)-31 b(yd)306 b(S.)g(\(1996\).)475 b(Semide\257nite)307 b(pro-)27267 68937 y(gramming.)495 b Fs(SIAM)395 b(R)-57 b(eview)p Fu(,)371 b(38\(1\):)495 b(49-95.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF