This is a multi-part message in MIME format. ---------------0506080537897 Content-Type: text/plain; name="05-204.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-204.keywords" String/M vacua, flux superpotential, critical point, Calabi-Yau moduli space, density of critical points, lattice points. ---------------0506080537897 Content-Type: application/postscript; name="StringMFVe.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="StringMFVe.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: StringMFVe.dvi %%CreationDate: Wed Jun 08 06:09:42 2005 %%Pages: 47 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMBX12 CMR10 CMCSC10 CMMI10 CMSY10 CMR7 MSBM7 CMR12 %%+ CMMI12 CMR8 CMMI8 CMTI10 CMR9 CMTI12 MSBM10 CMEX10 CMSY8 CMR6 CMMI6 %%+ CMSY6 MSAM10 CMBX10 CMTT10 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips.exe -P pdf -G0 -p1 -l60 StringMFVe %DVIPSParameters: dpi=8000, compressed %DVIPSSource: TeX output 2005.06.08:0609 %%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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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 49 /one put dup 53 /five put dup 54 /six put dup 56 /eight 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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7b9a1c0b2b67d77569ad5b78183fb786546539e668f4806bfbecd5941cd29e8a 72994f685fb99e4a8aceb73cc7b37b818476048e5da26aa329b1f52eb674bd76 2d701a21c8458d3b6a148ae48b97d5d15fe1fba1e8d8a2d25805b401093c7bf2 94821b5356bc743c684e0d191c88a766cbd0422a58e6674f0f2135a81fc7feba 7bf41c8361f0b7661fb109895bc20737f722ed0655ad4bbb03fa3146a3823a06 60bbf085f8a4a3ac7445c71a5d6ceada80222d26 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSAM10 %!PS-AdobeFont-1.1: MSAM10 2.1 %%CreationDate: 1993 Sep 17 09:05:00 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSAM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSAM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /square put readonly def /FontBBox{8 -463 1331 1003}readonly def /UniqueXX 5031981 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6f33457e3200f0935ea98f1a0cfaff49cfa2 d877ff805620cba84126dfd175a82e1b4c6ec58461fce97cdd749a012e2c4243 86b199a97c306b282350cbe5af90cf1d8e7371dcaedbd5fa3346dd3e558072c5 fd51a2edfee6890f62efd1b552699123f4f52c2817dadd5ece30b3bc178a3ef4 f45fd8ad0ee5bf30db5f9eee9ddefe249f3c349a886cf9130cba68ae559ad719 c439930ece1d52d38366116c390cbeabb8fdb36811cf3b8e2416825e8c880b24 6521ec6880ebe7f127c8741c756b29e1f89e3fd5d638a7bbade8c3a0fdbbf7ba ec37cd372afb2d9dd25d509726d20a05914e418f6b7e759c440beb61519df02a 06734ec1226890d17c51f25f775e6879fdc231bc4d7fcb4b33d04fa273bdb8e8 f124171714614f8a94e98a6c2d6bdc1254913999f9c181e42c0886556cdf48af 92a265c1eb06bb30fb2745840c17bd31c7b8bdfd37dfb9643a418a2eaae14094 6752e0a315207aef85c3c2630538907971ec1b05687208571a3504bb3369b5c8 9a43e5abe343d0599fd8d41a1bf01de49f5856bdd67d63a2b3ecea75d721a6ca 0c8751ab41e892e9cf674218dd44acfd40019bf59f4d69ea40d5cf9068a6ff96 9f8cc772b920baf17334833d732431a2969b9a30fe38bd44772cef653935a957 af61dd568f041847ccce3c587656cba7bbf46d8cefec936c6d34dbe51bc04929 0440354f3b29ed8e6b4f21ad53b5809779d67339f284563d54ed4d2a504b014d 974a31c0ceb57b364af519bad4e85c41d521a3ffc26dbac555b42dc191367152 1d60c30b95b5bbafb3c7d87a1c043ea5f0e6dab83a4de99a742e75f02837b923 c50eb478ac4131ee604e3d7b784bd8c93eb9430291d452c21be0495718546333 66a602a9f142a563fdae3c8d40de6a8aa581ff3172367ca325e6d23102c5c6a2 02cc99580287dbd76a05bb4cca591b12a9dd3d602d7957376ec30b16051a63c7 e26ddf009e21fdca5a31cc973f0b1f6967a66b30b973ddd9d9a3095b673ea8f8 5db8dbf3acb7bb2d85c04a0ea8853e4592e4f8e581adaadb78bc3ae58a2d4c80 3b09839eafc7bc3e7cdf784e852d6590ab1c4b7558e28505887c6858072f09dc 2caba91b6871920e5256b602a0de02b87e5eb456878d3634e8eb0e069fa2bb5c c10d5cd98401de1cf07ec9f7de62aaa20b583e2f7aad2fb7e727ebc43f712c74 e8eed1fa76133428ac71589b933cc37ad200c516045fc1a06d0086c541103b5d 4ec54dc77b 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM7 %!PS-AdobeFont-1.1: MSBM7 2.1 %%CreationDate: 1992 Oct 17 08:30:50 % 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 (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 82 /R put dup 90 /Z put dup 114 /integerdivide put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueXX 5032014 def currentdict end currentfile eexec b5e2457af4a6caaa4536453c498a618ecdce7b5f6f2d5944265309f833bd3d50 3df8bd14c071f404d2177500681a801856cb8c9c9cf0112c9bd17ccba9ab8944 1759a320b2078df0fafdd3e357994bd81740d1a7522ddfafb75d58f5def046f7 084d10fa2f702e570b09a7073d70efe37b48c355a7089772238b7b4538c4912f cf6703a14fa416057ea2e1b94455913dd271b42846ee10d78270a8118e7e4bcb 8deb048bb280c240d59409d47046c890e5f79e5dd4b0be65a28a77aeb53dd33e ac506a6da87bb55e329f433627e06c3c94b078838b908538d93f888e504e25e0 3a8ccf7d2c02122ab3d485525564295390d1154e15fee35edff4eb016fadd32c 26d1b4c09de87943780954cebc4965aa890fe1f80b03eb07103e13d468245c7b c26455f229e8859585c6eeafd9474acdb27290fd62950dc52f893c99a2a614d4 2a7e3618266f3fb0c68a01a30373e048385ffe7dd5844e208a55d1a9263ab2db cb9b19038724f95af00349840157be45cb337871b55a304b87ade64b10ff0eba 21aa0eaeef74aa01f049303a8b25de0acc26bb3a33fcf267f479a2a73f7dece3 2aa23083687322b417fe86e8f5afc2136d045ad6b392364811a17630a94ff097 038eedeedb8da4ebf8fa5a96ce27a2ebf771d14f4a11fb1f9f48cdd207209934 885ef9118463552984b0fd3cf15d55de967fbb051549dd65f60dfe07f299690e 58252f032bca78935b7d2df852ed688bd13dd171b91cf7bd46d5433aef9f50f1 41a2517c4912b837793f84ddcf1116f70f7e9b1e6ec752dbc9183d4791d92602 397bafe29ac6f879ceb72c673c8a1aa0135d6f92d0db7791f9a2f1a5b55347e0 3e63723efd258754f5f4a9ad35873a98d54b6c54b17a39892dc1d38660597448 fdba9258493a0db0d560c01d49f2d889a99eaa8a5b5919cbd33f6b3d97bf27b3 e0b94b48d5d21d6c29aee4faca71bd775bb9defa37ff627f1e4926f10d59313e 77d81a511027ef9a8ed5b7fa9b5bd67758cd13bab6ac18411c974a65d0c90085 67bf5860d0930d7f43e557ee45e76ecb2806d21f4732bb282ac8b591f2b8775d 4b3ce2a37ebb371cb74a3cc513bdd70a0115de4760c21f034f5080d769f7ff83 554af524184702a0294a070f8a91e20251ce976680fce3f8b65b3311e3030145 9b61d653566dde9c6e265c8d6378671c9aff736d3bb8fe263c2d16b3482cae3e 2bc52897fe5889d771321a10a6b0ccfc68291cc797a7a39052878ce03b519e59 9f0ca581fbb0a6ad041070f0b15d15c17fc2486bef1d4065b78c5b9f9954c364 b6b4258da91b9b6a103e4b3f4bae65f59246e671fd2107e9b7225da5c4ed2f29 477187cac236fb82c87b0fbf6de3315b877c8e82877d78be23e13b08ca278d44 225066b2eb5fef25dcaf82a9fd2ba821ff7fb0e76bf690d215ace3ea7943611b b7d61ca3c9d439b20d45f21855f50a611dafb4ec9a1e1ff48d7aac871a9673d1 db9dfe3fba3e9705bdc004eab3d275f045b322ecb0bb9a66079b28646c221cb7 c826ab7f6ded42fdbdc4635c159b8a2bf77c38a1834fa5a7bfc1f7856990b492 483837b582dd6712f3445d64fef631e846d5dc0967d2b7fc0e206f470b8862f3 809ba4aaf95f15b12b89b05170f61675cba61165a7d71171335bfc8ced170f56 3ed664ad63c3104a1a1d4599b64dc24ebc2144526cb25b8a927ebe0fb51b8ad9 e6f41b9b03b31685874400984a61b23a29467c062de432d4826beb14f3a90c44 3f7b2fa2b1363c68d52afd3f48ce33b932360089b2717bc8a934ee85bc3a2869 ab1a711b197676cff5ad52351bd46ce0e6cb3a61aa81c5dd1c2fbd1f2fdfe77d 366a8a0539e4c421f728ada8cc6f53ed0099981c30e0b158737c51634208a8ca cd34d439de52fb083ad71781558da191a4c45de7852e74a7cbb0d69b9eda0b28 a2570de0e0d45602d7597260cab183ef6b022b4d1131ebe33c03f82054e48f71 018c487d81e2374f16867163ecf8511ef23a1229501ab5289227943119596d93 31d57cfe97d3d4d53ed1dd9563b84fd700f5fb3eabab0650c586702d6088004e 2d942efc91de279e2e13616214761953a7b01ae082c7313a2333c9c5141e4ec4 da59fc7a33395efa389d5ffa3cca45e174e5df628dd10de9ca02c94977103ed8 60df152aa8a6b08349e3256310eea46abb71dc813d3b026557d9af6a351870df 4ab970ecda160d7304ef2cbd71ef18694e087a163f40c8d2a499d82ad4e65fed 006c4ea20f817fdc00ec2c965f68317634ccf41e568cb6210db893589f484c67 6147bfa9e68949fee1c49dfcf8fe8118961bf367cd55f3006de8765443530c98 10cda0f713c4edad31572f4950da611439be0e1b33ea7520cd53b4660476655b 1ca84f11199f9de41cd0ae68ee56a2c1bb58f3193144e4b9808ce646eec20567 032409ccbb99583dc20479282d5f4cf62c185e3656c9b209e18720d868565ea8 5a671c03c04a1f3163e7a3edcbdcbae8791102a1b8d3f762c13e952b49d0c530 a49ede26712fc4cfd3134e13dca9268bedb2c2c4 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 63 /perpendicular 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 c43e2851905abd26842613d8fb644512275d43f312eea72f7ea3ba90c8fda6ce 7a9dacf52f55f0dd557f8d866ed6f3d3d5e6fe7e9ee79404fbe1a285bc519ecd 7ca162f0729044aff526bfe554ed59dd9beab0f842d55be08caada6136ac8e30 21c0b1f2869eae249565a126093d019e874f1d3e3de66d54ad31a98634d992d1 2087aa9caadb695d492b62edb22e89fddf96357f7dfe18336e102a892d42c1a8 924b450df9e62931117872ba4c4f60217f8dba3846b1df7f87862b42db5627bd 43eb0e86fd0a633d67f9c8d2e64df360c689be9b696536fd7d1886be07beb475 10befde02910376810d3894b6c3e16d0694dff896b2cd98f5e44f7f059776d8e 9bb6e3c852781ef8e6e0e0a9c8c78b899754aab4f9a8f9da4fd51bc78e39e7c9 a538ff680ef2c545bf9fcb4b038ed25337f8c02f345b1b7100eb6a13cac6a3b0 9cf4364bbc96c87ee2b6838e4c6fa5661867a579f52f3ca1a2dd9cdcb8d12bde 349af6761ca222ad7bf07c823f98bc71f70f28b29ca150c66eade8936ce71e02 4d8695ce5a01baccc7f507808e8d9b6467af00b88dc2f56cd096d10952aa53bb 0b06a6df6ad82ef8e45c90047d4a6416b6e5978f25eac8afa82b5663dad7cc7d ed3206bf5957291df837535db23ba63f9f7aca7141e1490a68327e35fb7888c1 60c2d47bc4a7cd84194ff52646df43ac83a51489481cba4d201e5094e7ac3ee6 6a5828bf1d87a530d7786577f164ac3d5c0d624fc6cf1ddfffc2 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 22 /mu put dup 28 /tau put dup 32 /psi put dup 34 /epsilon put dup 59 /comma put dup 71 /G put dup 74 /J put dup 76 /L put dup 78 /N put dup 80 /P put dup 81 /Q put dup 84 /T put dup 87 /W put dup 88 /X put dup 90 /Z put dup 97 /a put dup 98 /b put dup 105 /i put dup 106 /j put dup 107 /k put dup 110 /n put dup 112 /p put dup 116 /t put dup 117 /u put dup 122 /z put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueXX 5087381 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 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858e6ae1bbaac7a8a5e111a401de652f66466accb192d5066323fdce97123ac5 31800a1003f606e6b218e6555ddde4a962cb8e2291bbfe7660c95740dff40986 9a27175390a398055e91026b2aecbd88141d7ff5f5f48959c401939bf43975e3 44082a7fa3abaa11c0d35856ce9227ffc494c5f06cfdff42796ad38b507431ff 9d93bd17ff908d98e9fd19d05c338ec91ef0287f7e50f9ddfa48abb9ac608337 d1734d7c89a41d0a0148efaaaf7b4fc46df72e83ca68bb5a3f68787e5935c8b8 47c30d48 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 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 readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueXX 5000789 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486858b2a4b5a0cc2e1bf4e2a4a0e748483c3bcf 5de47cc5260a3a967cac70a7a35b88b54315191d0423b4065c7a432987938c6b edad3b72ad63c2918b6e5a2017457e0d4ebc204b031f3fc6c13d7da7277a94ba 018e9998b3dd888011a5d7c4204989f30f908b95533bda845746b673ab71ea57 65a0d14f4350707e47c8276305b28513cbe1bb0dbd269a53719bda46e536685d df78ca0146b6b93e760256b74d939d4e35b5e77238f04c92298dfdd188feea30 e053eefbcbb52f2011772b3aae39f5805597bbc1e8bb75a446ce014030f4f2f0 f49f9e962ee4a1024a746fa92a3628db5270732b54e43fe5ecfa524f127e5fcc 788e77e66098336ad67fe4cccaf0253272d5df79864bf4b734cb9a5859d557d8 bc11b8e00221ebc12e97de4b1f466ead83a4c894709363bca9040410a52d592e 34ee40cc7e5efa920546b981aa659513a24b1b85c221a1875b62d0b89e57a368 321b8043a5b094e0379760a443d632892b14ad6d19dacc8c78093243ad67e6a3 08e56e6b68412ee690b10dac6e17708754a00d51fc957b500eb80175716eef4b 2ca1ef867614659bee3f2b7319e97b6fdf1efc847bf3cee3156f72f21751da8e 5fb6898919e6799820d3de0642d756e09d6fae4ff08dd3deda3173bff4bb11f7 9109c97ddc05897af709ea199a90fcee8ce4c7a3c15b18170c41c04de2d3fba8 f34296a95b8e1e8de3739b17273f8f2c85e914615e8eac5e8bd2387ba3b1edf4 7968f06e2067d836d0f9f3e085cdfd2de06a62c81d786b304326f7002e83160a 36598589228b4dddddc43c85e1d126f8fe81b828028e26317af5894aaccf4f69 6301e1a9fc45935d8a414957f08febebbc3a72ada80f101e47447d019ade56e9 f4fab969bba2b44e47399fedf5caa1bcea216d7ba713d523d555ffdd58e0247f 9085804a8a2f6c15a1af12e7cf9c18b4d9db2925e879a8efbada0cdfe3ca06dc 89a4fc0438d139d125b8efba730a4ba26c027ca39b586d7062b8f8acde32adde 6f50e665daf59de4ca68f31d1cc73ad988390730b5298c66475dbb4e3c176877 875fad04433188d4c7d6117605d7d37606c42d365245ca7f4e7be0d4c4e21dce 7073bcc0d77d9109d959921b1f1382be25ad206f2ca6a1360d92c8b761c3f037 1b656535280ffe7879ab2eecfcf351989a513f50ad87db321c2efbd445ead8f9 faab9f9772e3c4cee2c8890072b0a85597fa7e8e155d4b2f61d2fd3a95cf695f c99e62fbc96af1ec728b471ecfc953bee964bdf626ee4fb91f3f42e461ad24db cb6c5b4054e29678001db9ca4f6d470f925ad11977b9aab454ad6b89251e38a8 8b0775f6e3d84974c65f965c1572fe2b891f728346b8479e32d47379ac1a27c6 b34c83cfc3685f22160a1abb4c5fe7cc0d25a2b1cb7228afe9ce1a40485a6190 d04448e62787a8f17bec0df2a546d5224627d70d035539123e62fa94e0783308 b14dc92983f87c9f1a9d3ea81c79458c4a2876d714b00777f84a32fd8adab724 8c6c8d630528001f64773741c6d7cc21ae2e8aab84db9d409411d948f02d3304 12d5e532dc74132865f7ccb7909a60f67feddaf0f30fa1e4cde2cd51aa9ca693 03921f399998d043e70e0ae7d86843afbf3cdcc30323d3cd4486f2f0edbd24bf ec7b8ab911418305e3cf7d81a31d74d4b645a009626a0145f23ca8569328de4a de89e51e05d269d5cb3962a685ab8e42bb541fb1d5470a3f1b79290252626a4a 4d7165e9992800badbc3ac742ce74779190c396b7ca6a9c6965cc12d626a3dbf eb16c259a9d2375f8bce119a963b43733ae30edf41b902defc7da2337370c03c 4ad3397975482c9c20a88fb40ebbfd4596d6bd10616bf5b4a41b5a7fa765a3b1 33e9d458241f3c5cda2fd57c4c46d5cf0bda8a8f4855da182cfbb3198b781ffe 81ad160af0e146eced26da00c822efeceea2ff756da54b98b7bee16cb98b3319 7c1592e41e3b54aae60dbd2c3fa48f77a2c80d9c15840f223cc9f4b30b63a963 1b54c94859e3d03abc81c8dd28cadc2b025f2012908afc22a24942f17817b7a0 938d3513de3672c882e3b17062401603b850fb241cf4650e82032e547b19bdd0 8f8e601b114fae8d38a0c82794dd0eada6438d157b121addd40df5d875b89c16 f54e8114b3ebc2fac30f018b7e4de91c2a9b6d6226af308565104bdfad9a7248 1889716c73c78df3a21e08e501e0e662d443efeba482782ab14b4eb523013dae 9079218d789d3ca9de3efe6115f2bb3673c4d32acc989dc689ea92146e8219b4 22aa0d44672da3e76f876d3a2e7e26c72348de745fe538e2ca3ae679c2700d7b 19cc0736e4e30366330cf3edf3d180d28a0f2c1fd7568f00ef7e4ad417ed28fa a1968e58d46becf00f15e835eb05cdaea42c3457fa775d495abb0c7ef9b65b5c 2888351b5e3b02103453e99f66da9e029afd640bf0a475726affbd95953d0d46 2fa14a07 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 2 /multiply put dup 3 /asteriskmath put dup 6 /plusminus put dup 8 /circleplus put dup 10 /circlemultiply put dup 14 /openbullet put dup 20 /lessequal put dup 25 /approxequal put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 63 /perpendicular put dup 67 /C put dup 69 /E put dup 72 /H put dup 73 /I put dup 76 /L put dup 77 /M put dup 82 /R put dup 83 /S put dup 87 /W put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 112 /radical put dup 114 /nabla put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueXX 5000818 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b66566ca5ba42b1911a 5d7f1bf343015eece988b7a93bce0c7aa61344d48aed9c92c8698d4b7c9951c8 7d103f2414b39e1437f9d2e50c4ee5f218f2e6716926a79ea978f13b1f855345 191dd7d31d8f82c2e3343c7a5894d95bdc492c28226834efcb5c12fea36ac5cc 430e0aa604961e34888adf6c1f3954cbc2498e225d953cf5685852162346f474 5a2a7087d5d7ad486de16d2ca8e15cee26e012671ba3bdc7d95cc8c98bb774f5 08625e968aee27ff7d1a06e63bcfb5aa4876c3f8f13b30ccccee73c3caf4e70d 98e6ed2f422dbb4950bf789680e064150995941a9f4dd68a575949847a7d012b b910bf03a681f0c4c84b3054a623f973471de50e2254753ada995f3e7ce33d8a 7d01b39f0e5a9675d7c24920d117dea104893f987390bb84e1ef04bca7664aa2 dbc5766a2ee105ecd91463bd60228cdaf8e6c43008441ae86ffb26f7762560f7 65f0a47a1a2d0083b6fd8728cf67234ee1cac1b762ef073c019541eba802e4c8 76054916132386c0f152952d31bf1d8e977e6cd3b496d35a21fe6500ca497d20 9b84068b9418c14275777bf42ebe4f8705ee6a34a8e9c5fb3f6e1fe98c622070 1ff3e56c68df6473710f44f93073c3332ff7c7527808f207ee8ac51783e5e3b8 21f4196439b66a13bf7497c58aa7e57eabc37066c1639eefa27fb0211da35674 5ba0b21439cab6f58945748840ba3a7a50ec5f66e30e8de47d5afc9daee91365 d9d92ed80a1c9aba25379959d60e3b678a4395dd6b495b7f60fbb56bc64e3fc4 a8c11a41e4696f425d969e05d7fdd267732a630aa78dfa742467d7eb5ffa7136 644bcf82d9105597dde87ab3c281450ab640c239807b39c5296a65c984988944 b8802321f4d74a33c32a2f59dd74fb0348a337195ebb040549f0df537405c5b6 56d3b45e7dff2f0a0072b58ff86fa023d019ae0b9a9eda22c6f9134d180c074e 248a837f32b3de26e7e6fd5051a546d5270dcfc0903dcb7100b577bd7643189d c258e70bd08df20f3094f177270185a835af825b021ee663bf8f914a9b229a72 16b370de3417abf5a269a7756531735c3f8fa644aba86455c0eefecf95e65f92 5e242760edc20a3db8f59d8e279c280f1ccdaecd3f38e7b41a17919387f46e57 a0c4dc2307f0dd956501793c2e0af2e19341c82f74246c4a32af116e780ac4ae a134e2f8cf95e68fef17d251b1544b6313ccfb64203e8776e9ff1c78e84d5419 376c27cc5b42ca04567bfd34b373d78111133a5f7ae0a08c216a9c36b9426dbd 68ed3cb9285663a389dcb28bdd8a803a2593ec84bd9e8bb4583e36f40e40ca78 b3abf42e33f28998858fa3bb4a84187e32108317f2cdac414f9df70b6a512370 07d134b5f76d8f438989252e3895d68405dc7ccff48d05369cfacd6533da1923 b417e102b306a7c676b754783cf69f59985ab5b6183d5e66e694671b1259c56b 43c4cf8d0cf994351c234cbb9ccac396df843d0678fbb591ae9f49fbe1c91bdd 036f162ff81c07ad50d04d4f214b3c7e1bb897fd351b210ba9f818a08e2ea461 101c156f9bc88c996ceeac91499c32f93013971c18ecad28b4eba655fa56cb0c ee9a46115ebedec22d3a8565653dee9125bbbe693596b35b714e6933f2e7774f 3ea9340197f20a9b04e98ceaab95281581f8bd954602b82d9171afce667c38ee ddd3cbe4124f43c9703fcbec7dbddb845e80f8d372eddddaa7bae37450790802 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0d398c3d481e2d5fd9a7fa8b48c1fda6a9c80e89e2095a80ea21d2b13730a988 c5b579bfeb1f6e0b8cde97647f36f9d8e90be83f3fa50e15ecdfdbea3a40c799 384a81226e59d9b31493e94f989b76b8c0240b6852fb2e8ff68f07b94833860e dffbb5bb6cbd3e80e8136974d5c430ceda97b7bffd36228b8c469eadcbb8d500 bd7942d303bde09fe96f3b0aa311742a81f7585a95fe91600fffe7e18c3045b9 d65915b139f404ad06785a1b6185174a17c14c6ce30ff955ceea8bca871791f1 ef4dbfb53ff371db71d598968ac3490672af30c03eb27bcbe56ab1f2da520b53 2036375942f38efd51f21a4e775a721815e47e20ff1b0713e15c3769df0a4630 621864502a31242689486f0200ebe163c54854e7261f256e88a614564663380d 1ec1a16f19130bfb79964df9f9ec0e73129217663315cc97a4495507a49815fd 1d14d3a6d66576baaeba28c604eda58ba19c317b5ecc095cf1e4c9a88f978924 40c9a4e6431f640154dfc684bc8a2312b37523239807d0fbc91521786f6261b5 834794ab0767271750869646e4a90cfab8fb48ed70eec7640a327f9a78abdd65 9afdf264cdb165fd521f2f3d09c542efa6d6188f1c6ffd26a3e00381a071179a e69fd1aa5fd416e839cf32adc63af3ba8da2a02555637de182939e233c7d7d46 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 2 /bracketleftbig put dup 3 /bracketrightbig put dup 8 /braceleftbig put dup 9 /bracerightbig put dup 12 /vextendsingle put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 20 /bracketleftbigg put dup 21 /bracketrightbigg put dup 26 /braceleftbigg put dup 27 /bracerightbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 56 /bracelefttp put dup 57 /bracerighttp put dup 58 /braceleftbt put dup 59 /bracerightbt put dup 60 /braceleftmid put dup 61 /bracerightmid put dup 62 /braceex put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 80 /summationtext put dup 82 /integraltext put dup 83 /uniontext put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 91 /uniondisplay put dup 98 /hatwide put dup 99 /hatwider put dup 101 /tildewide put dup 102 /tildewider put dup 113 /radicalBig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueXX 5000774 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d5649664203ab24acb938d58d246707ffe7 d62f04bec4b70c21ef75beb2b812622b3c74e969d72d3cd11bd7106294a99caf 0b1629bc7d4de6b96ca82930831d64575f23f4ad06a0e45e315b1d392411be8d 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18361ddda9213b75c4015d27520851de1064c8c8789ce93bb6fde7d2d6e7656a 3c6a49eaecfc953fcac8b5444f217e3208463f5498a6147f6e89730543077be4 a4bc03890591cbf32a5a78a6d30f77f2ffc81d76ffd072df44a14577953a08bc 6dacd358e75d09eb83d26beda61007226e5bc41d8bf842c450efaf077b1caeda 84fda6abab7795fc4811047a08153d811dcf945c5b54fe8a20adee426220da28 85d5799f29509214837bf7171cf5e82da8e00fc703f7cfbc6ba2082256894995 8f2a59a2660126f251f65eea88edb81685480ef78956139f2743f6b00d293a73 9854cf129b5e15729dda495b63f2ac225e71d11185c0f9d0ef7d222455681971 6a5b10840254e6e68eafbecb62a8d0859d9232502561e3b7b9019e44bb091b67 c591c532c0da254c779d2b9fa29811bc07e715e21d62b70c3fb86959f92f3cce 93b0fe1dac341906893507418654a6c17fd6d5d010626d3e7ef839f83800f60d d33d08372c36a85103a2c276afac0f3f49694d2960c63d76992d6a58f63a4667 1518cc6ac2f72d8659676bfb5d6d67b2eb3bdcd1567edcd4ac6f8b965bcbfb30 f7cf969a8e12719d6c072d0576a71cbc70af7eb98a5ab221bf8365bf05767d60 36bb9289d3e83cd3131df6749514a0783aa2d420b27628a78602b9681ea02953 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1b4a05371873c47d425d1c89fd44a54e10e0cdaa5ed844b88721aac28304f7bd 76dbc3dc9189477f1db7b8bbcce55d86d6a48e493115ac5c73f2335923481c4f 7e718df22bcca5e4818a5e55e396f00df640e6d0426437e02eae111abb1274a3 04175309098d014b13db025da3983ec15b6977ebec4ea09f61497424b19b6f0e ae37960322e5bc1e1cabd8d5a84360dcba805ddc1d8025f89bb63e334784ebd4 7d1971b2bdc1525878cc2c6bf75e47d8c466c65aa3d1325f093d78bc574a638a 2527157f1e28dd0227d4163bf25b3c83e577e65a5c90479bd2f60f710d6da60a d6ef7f92c6938ed25b3fff86b6a495c2d3e08a7056a79e56f7e2844a87228ac7 bbfeb936df8fd625846e8b0ba1ac989efc45bc2e79adb3078283ebe2d3a08f0e bba113e1567e4c8b5f818974ce8ef91859127946f2fd182bddc19a7c99863ec3 7d05fcfe6401e1fd7d5a30b368bbdb21e864e54cb8fb1eb6577fb56e5497c61d ac24ae50657fdcce2da8 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 80 /P put dup 82 /R put dup 90 /Z put dup 114 /integerdivide 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 a45b5765f0c1368c063d3096b53905b08d42e615e83553eb3622d12f305b9a8b 3900e3309336c2d476891babdd1f9dcb3848c6aa4234cdf23bc38777ae9da891 df61515221c95b82f6ee413ed85f52b6f5bb68976980cd6cbbde7ad6cc467d10 7bed768d1983c04f6f12527de491c34502aaf8fd3102f6d76999ce73b532c596 b35349841bb2e7715db2cd3817c0dea12a7f0e566cd841667ea0e57727bc5516 9037fd266e86bf21367d3fd1101487ab643425eb707aa4d28b3d2e4b89049e30 4e02c8594aa3064b6581ca20073acb066380b6ba4a394a016591321852022194 c47a8d50653e6df0b557ce0f5b1a6141ad5ea2187547277022b723a4ddd00d93 2ac5770eb10c2c67871936c14ce937d9151bee63008dd335b13ad0b24045a6aa 671b3c4a95791e22d59d6f4368006f527ddad13680938596d090151ff3315f14 44d3de5e995f0b53ee642db6f3222b97bd1ad60646cedaa3ee129abb07cc9c07 bc718c747a680c2f8f0cb1106459c2299dd72da33b7eeb8bbafef048b339f602 373fe7158511d87c61831635dfb64d0bda0782e29684ec1865c488b6be05cac6 d3d7ea1fc5de265aeaf238bc4022bfe24ec4fe5505ade09e5b223e747182140f 24712e60000b1df56ba9fbd533696fb8977b30109b4438b6c9bd13431ad516bf 091b35269c94b22e207d2083a54fdca373528fc2b2567039cbc5064b1e08f840 03699fa6b92a84edfb83c138a2967921841a7632817fda104c1b962341a519c1 77e571937361e5b6fb789b459f1ea15350c19e22a9eed18c3edb35665e537f0a 2c5c9ab4dc4102cda7fa25d00eb4522e71732a8a657fce9c10b70f713b098243 fcd8d5b5e10ceffa5a7a5e5cd6a911318a01ca621abef80f3b4e0e8a5bb56309 8dbf82fcd8841e068436ad923db5c2099e10807fe6a80aac5f9aba4be0fd168b b4033e02d24110e3e3db99d0328d9c24e3e37c30e7e496a6f95601de4bfa6e2e 94d574ea2bde51212f24949e1239d8465587422edb2d8dc22fee0f5c9ce73ab2 738aa367ced895e7568f03b31b4168cf85cdde2fd74b6b2b35712b87f0e3a27d b4826348b9a821d36fbf341adcd02b488e65bea905970b2588fb0ae67f643f0f e127408efd92bfc475d89fbc67a0360cae673c4f29b7637bfc36d02aae459e59 928fd10bc5d612618c28894a1d2f7bd8e2ddead5595555febc8e59051ba11ff8 81b4aa0222f743c7785afcad37df8689aa07744a4c8aaab7857773a9c97ca115 8397f772f21842bcb85c3af4b5d446eb8bfae65162699affec8de5c1174aebf8 9bee69924e99cd6105035733d6cea153235e50736ddea7209deb3a74a5d0fc24 6c49c92df5ec4635986e33314026ced33fde09368acc6b35759ff6cc25eb7a9f 17bffd67982602e9b293d7d1f70ffadb564f632124795df98e3ef72b807719b6 a217c0696f00fbd22801bc43ca1805247af6cb7b17af8b554e491b9889c8098b c26aa8968335e2d88e22dc5d8f374480e13d0910ccf0660e6c061cbac73d306b fc97afb553ab467c83c2ded6cd4eae0950387726d86918e0416c062962b25caa bd8404f7441496b864565bda313f995afa430402642cc87c98453f8805c13a27 12359e9734c870c31c6e4aea165f636597eb2b7b7281b1c2872ae4203584a98d 0d6eb7e614d2982cdcca81447705bf515eea2093acac8cf20ffa18da75a7fe07 2c20060eb85f54b5e7e6c375fb95b8d2c46a05f534f3b9ea30fb5d1b1397fb47 06bdcd96 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI12 %!PS-AdobeFont-1.1: CMTI12 1.0 %%CreationDate: 1991 Aug 18 21:06:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 58 /colon put dup 63 /question put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 72 /H put dup 73 /I put dup 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 86 /V put dup 87 /W 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 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 dup 127 /dieresis put readonly def /FontBBox{-36 -251 1103 750}readonly def /UniqueXX 5000829 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 66aad332fe6fbcced1417802822a3b81d6187875263d6bbda04bbcf6e4870fee ad60f104bb3c5a766610dd11aea64a6b107b0b04439fa2888b8cc39232bb83f7 695aba81f0260cd5248b9e649cd803271dc8bb1656323089e9e2bb50f2b95088 87d31dc36e555668d9578b338402f8c259f6813b0b71e6cbd95e19c20ced8495 2559e47577e612a4d94de3c1b4c6c41a6a3fa9e2313457242de74c7e4c0d722b 3e224d18baa726369542682e9459b951b2ac638494880c42be4e2175a3c067e5 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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 44 /comma put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 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 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 89 /Y put dup 90 /Z put readonly def /FontBBox{-39 -250 1036 750}readonly def /UniqueXX 5000792 def currentdict end currentfile eexec 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 39 /quoteright put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 57 /nine put dup 58 /colon put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 73 /I put dup 77 /M put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T 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 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 121 /y put readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueXX 5000828 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3ffd20326947b1065649b533eb5e9e3 a88a87fe987918a687950b26f647d6e08bf14d983ed1d4a8ca1e2da8d985f944 2eb916f5b6ed0697ac7c33e1e36a3bf460d34ce45f1631871097cb04f18e3889 4cf4ac1538eb19481311d24fe3be7beaa4a3730e8b4831fe59d6d9ce2e46116b 629c7ba2f9ce3ecba2f43bc162a5a077ca1b2882a42afdcec3f4b75b5d63e0bc 8e5dc95257766d8ea467ad9cbaf47be60f797580cced6884b3a68f70c91f4fdf 80fd00ed9139e7f480dc3a76af72ad9b434187730bdfaefe4cbfe5c7edcaaf24 9204fd703011932e5a3c27be468b7dab69daa18dbbb6335ab8ddfdc607961c7c 02582763fa069d43563ba17704029945cf42fcb19cf78c51df0ec4d851086d43 c1ab38e865ef36865fff3a08b01cbcb070ba4893f7482dc7819ad03d337e520e 7d8cd83bb8ac7aff4df36751f4f12aaaea2b6d7260c09a26e987c904003e0723 7dbcc4a8e4e85cd3259f40d3caf55c742cbb40ad0db1044c20a4f122a63f7b81 1d945bfa69662189ce4d089464d2ead9f6ed60b581b04114d5d45e97be27a576 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acc2e9bd83b712216125bd914a 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 22 /mu put dup 23 /nu put dup 25 /pi put dup 26 /rho put dup 28 /tau put dup 32 /psi put dup 34 /epsilon put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 71 /G put dup 72 /H put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 81 /Q put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueXX 5087383 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 dad7b4e6d6f9cf0688b69ba21193bf1495807e7a196cf14c95a4e02f9cd2da8c db2546c6df52e524745992e18d9ff87aa25e4e1800bbe4ebb357c6ef55ed6d03 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fa3b1c10694ad3e8370f06358a50496ee9d66ab7ffa765627fbb2766417761c8 34b53b75610889bce3c8f7b2911e2a2f6f1faeb067a44d1254338f17f8dee793 d3a9d51ad7c18f16b3b66aea11450f9758736a465f989e9f833c9716003f4a9d 41a03e8159131fec97cd46322093498d44d864cb1435dbb312726e835e298af8 65ed47d4ce77a815c939cec3495bdd79ff3636214c11c5cf7f74128234c2a080 5587659255c9ccdcba10ef 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 3 /Lambda put dup 10 /Omega put dup 22 /macron put dup 33 /exclam 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 52 /four put dup 53 /five put dup 58 /colon put dup 61 /equal put dup 83 /S put dup 91 /bracketleft put dup 93 /bracketright put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 105 /i put dup 109 /m put dup 110 /n put dup 112 /p put dup 114 /r put dup 116 /t put dup 117 /u put dup 120 /x put dup 121 /y put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueXX 5000791 def currentdict end 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 6 /plusminus put dup 8 /circleplus put dup 10 /circlemultiply put dup 14 /openbullet put dup 15 /bullet put dup 17 /equivalence put dup 20 /lessequal put dup 21 /greaterequal put dup 24 /similar put dup 25 /approxequal put dup 26 /propersubset put dup 27 /propersuperset put dup 29 /greatermuch put dup 33 /arrowright put dup 35 /arrowdown put dup 38 /arrowsoutheast put dup 39 /similarequal put dup 40 /arrowdblleft put dup 41 /arrowdblright put dup 46 /arrowsouthwest put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 57 /existential put dup 59 /emptyset put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 72 /H put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 82 /R put dup 83 /S put dup 84 /T put dup 87 /W put dup 91 /union put dup 92 /intersection put dup 94 /logicaland 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 dup 114 /nabla put dup 120 /section put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueXX 5000820 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 16 /zeta put dup 17 /eta put dup 18 /theta put dup 19 /iota put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-30 -250 1026 750}readonly def /UniqueXX 5087386 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 2 /Theta put dup 3 /Lambda put dup 5 /Pi put dup 6 /Sigma put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 19 /acute put dup 22 /macron put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign 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 63 /question put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 96 /quoteleft 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 126 /tilde put dup 127 /dieresis put readonly def /FontBBox{-34 -251 988 750}readonly def /UniqueXX 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9e37457775dacfa4e61d9b64eab180067a38b0cd41c270c8184be6f1457a6063 4dd38f94966256c4d113529c166ed3b2bd431cc48d2e 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: 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 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 72 /H put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U 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 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 dup 123 /endash 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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f77ac3d09229fd98046104cf1e37669cf9a7198f3b27feae5a3a067db1138edf 3c2843abc316736a52766c0221e284501dd06a7617f722af999a4888b8627d36 17181a956f67ec829fbb68f0372586916f3b43811b22dfca601526 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 48 /zero put dup 50 /two put dup 51 /three put readonly def /FontBBox{-27 -250 1122 750}readonly def /UniqueXX 5000790 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b448685940729e6e22f4be0e08c6505cba868f7950 93f556b64b808dd710eb936d3ac83e5783b5f7e56d363a59a026a5619a357d21 c4e77bea121eb24b09027d1aad93158edf912940e91cd2f17922b35885894644 7773730bb19af9e70453e1b78d5edc123b273a979891d6d5e624f4a388f3897e 13b045e34b4ee2f5e4fc5e996d3549640010c671de14d18b0b3cd300e57dac03 22e59bfbf7b29422230870f5897fcfaad4b50c7c1c58edcd119218163d6f6e56 25ccb882db7b6d09a1586508e8cb642a11c29a46915e4a96e282079cb21922c1 c2e360b487a45302fd22ec8c5fab63e54b5e844d4b17ca2fff37c69c366dd72a d02922c14c0932f65ed03e4219c117962edbad2dcdeaa9c10ce8af38a4ae52e2 b377245b0be19a77d6c936e7530cc4d0b78d0cc4a92698fa2870fa54f2d8503e 2d17b3d52fb2febb09f2b2af0c2a1892039ebe19a690098799a858e3d39631bd 6925a154d161df3918074ada6bd52baddd0adc3f07e2d9f15e27cbf7fe8b98c4 07205c811121fa91e059f2f99322fed63f359ac9da97aec383f067f23e5de331 51e80f0a88ab50fe8fdae4a5de93c1ae2fdca06150b37246140c0e87cb2325a6 0d2349162ae3ac93144eee1e665a1289105318fdfe86b6e76251cb25adc967d3 d0b97fe5e279e1161736ab22b4ca510b964342383a840defd38f96a7280e6ac1 34e48d740607ff2e7804164a16d47735864db847c97335e6d4215cb99911a1ec 015a3edaac1f28fedd56d2467130d07bae9416c15f0827d27c6c79f59054282a 418c12c157c91223a829947f47592f7cafd93ca182b25a73a9419127e3b12a9e 5167ac3963f2b019b338ac46d63880f94dda4b538835884d2a5538c85528d6bd 977f844d32b43b0e48caed5a4bdabcefe71695d69ca784da87b5d0b13f35180e 58efc5cf12de78792a88664ae1eef40d0341c09c59c2f1e4a8e79ba39cadab2f ff14f445e5e0e76dc03656120a1996ba1cf40916166d6d59410449dbc76c70ff 6ca4958be2a4f019a107e9d75df299a5d93866be8bde946a4b2f101a1693f692 49ad4793735248e2c9638f1e8909e314ef770a86430f05eed75613f753e94e47 e10b780afb13880fd50d274cc760ac7b05dcac4debcf886c7a4d3db15042390c 1c42072be174815a53d3c34658cfb7188e45bc3db518cab4c93ba7471cb17da8 bd65a3279c3323fcfc9e318a0be0934d99eec92495f172146cafd6017eb4966b dab7a2cfc995f53d68321bedfc3154c0a9f93c99ea8f66f361a3998704b26321 a9e262b41cc7c333aa9fa99e57d71727792ce3774d826ddcd09fe0d49670780c db6e96b615f76f062935d5395be9d8c398bf490ea5fdc2b1f58b1bacb8c6428a 3014e87fc55b6b7165b47bdb25938bcea78766522dca078bd396c3c3c638b1d6 c8b5f639a7087e49559bab2b5831405c02087ca38715a3982b655888db7b4389 0bc1ea86e25d037edc5167685a73bc19e5c46f399371c8e9a8e17c5b924e3fc5 fdccf4c79ebe6988ab709abbf9b2616b502933ac7a3336328b8f27c529c59dbb e77cb5dbe360664d8a0c5f51c67c58818c02a9e461a21378390487ae96504175 3a70d2bb3b9dfddc01ee192001fcdfe47176b8629679746d73d507553096468b 89cb18633db9a5271aae398babe6cb35db0a077f2df0246846cdc88156a36af1 f99a1e0e8f72732db23fac02ef1f6864015566a235ba7cad1cb1c65477ef337f c65e625dc478a2a57ca60a27ad6cce7d6e93ee0a3c14ee3a50af516c8b747e9b 1c17e8294c861c2013e47eb50c5662c48da3d34e42e0b0477ce96e6462dc71d9 5da2 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 59 /comma put dup 72 /H put dup 84 /T put dup 88 /X put dup 98 /b put dup 109 /m put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueXX 5087385 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930da4f8d58abefd45db119b10eb409dd89792 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 19 /acute put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk 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 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 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 127 /dieresis put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueXX 5000793 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 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8f27cd43488eb459523311d08c0ff3fc88d2f224d048e98b243fb775855d958a b73fa286b8363f95957e45f12ed2e06e21ed6b5a6ae19594dd667d1fff5f52d1 5f53489f0b1ad8f27e78356d04c118e90d303451bd9014c9a5bbd8bf4756a9da 33e0391289950922028c188b53bf42e1de59b9e984a7f9cd1cd75bb21f2c10db 7286f7a4e8d668451fd97e225c5b780b11e9ed9ebbc2087fb44ceda8fc6acdc8 eef83d72ac0735bd2d253f4ffe6e81c8d2cac2516080ef24e7c7e0569ab60215 524ceee080946e0fe10b280f0e3756e0ac4c817e626b637cc054 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash 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 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V 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 104 /h put dup 105 /i put dup 107 /k put dup 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68589 y Fr( )48 b Fs(\()p Fr(Z)23 b(;)221 b(r)36 b(W)181 b Fs(\))368 b(=)h Fr(r)19525 68040 y Fo(\013)20185 68589 y Fr( )48 b Fs(\()p Fr(Z)23 b(;)221 b(W)181 b Fs(\))p Fr(;)2822 b Fs(\()p Fr(Z)23 b(;)221 b(W)181 b Fs(\))370 b Fq(2)e(I)98 b Fr(;)656 b(r)405 b Fq(2)369 b Fk(R)38761 68040 y Fp(+)39917 68589 y Fr(:)0 70681 y Fs(W)-108 b(e)485 b(consider)g(homogeneous)h(functions)f(since)g(they)g(include)g(\(smo) 36 b(othed\))485 b(c)-36 b(haracteristic)485 b(functions)0 72231 y(as)434 b(w)-36 b(ell)434 b(as)g(the)f(cosmological)k(constan) -36 b(t)433 b(\(whic)-36 b(h)433 b(is)h(homogeneous)f(of)i(degree)e (2\).)p eop %%Page: 8 8 8 7 bop 0 681 a Fm(8)7340 b(MICHAEL)340 b(R.)i(DOUGLAS,)g(BERNARD)e (SHIFFMAN,)i(AND)f(STEVE)f(ZELDITCH)0 3337 y Ft(Theorem)506 b FA(1.4.)558 b Fl(L)-66 b(et)471 b Fr( )429 b Fq(2)381 b(C)14933 2855 y Fh(1)15930 3337 y Fs(\()p Fq(I)98 b Fs(\))472 b Fl(b)-66 b(e)471 b(homo)-66 b(gene)g(ous)471 b(of)g(or)-66 b(der)471 b Fr(\013)391 b Fq(\025)382 b Fs(0)472 b 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Fg(3)p 46987 7408 2258 40 v 46987 7956 a(2)p Ff(b)47757 8155 y Fg(3)48218 7956 y(+1)49433 6824 y Fj(\021)50226 6426 y(\025)51149 8299 y Fr(:)1328 12048 y Fs(It)340 b(is)h(reasonable)f(to)g(mak)-36 b(e)340 b(the)g(assumption)f(Supp)220 b Fr( )152 b Fq(\\)29231 11712 y Fj(e)28960 12048 y Fq(D)404 b Fs(=)369 b Fq(;)p Fs(,)359 b(b)36 b(ecause)340 b(degenerate)f(critical)i(p)36 b(oin)-36 b(ts)0 13598 y(cannot)510 b(b)36 b(e)510 b(ph)-36 b(ysically)512 b(acceptable)e(v)-72 b(acua)511 b(in)f(string/M)h (theory)-108 b(.)808 b(Indeed,)529 b(the)510 b(Hessian)g(of)h Fr(W)691 b Fs(at)511 b(a)0 15147 y(critical)575 b(p)36 b(oin)-36 b(t)574 b(de\014nes)g(the)g(`fermionic)h(mass)g(matrix')g(of) h(the)d(theory)-108 b(,)610 b(and)574 b(a)h(degenerate)f(critical)0 16697 y(p)36 b(oin)-36 b(t)426 b(w)-36 b(ould)426 b(giv)-36 b(e)427 b(rise)f(to)g(massless)h(fermions)f(whic)-36 b(h)426 b(are)g(not)g(observ)-36 b(ed)425 b(in)h(ph)-36 b(ysics.)576 b(\(See)425 b([WB)q(])h(for)0 18247 y(de\014nitions)433 b(of)h(the)f(mass)h(matrix.\))1328 19985 y(Let)j(us)f(note)g(some)h(k) -36 b(ey)437 b(features)g(of)h(the)e(geometry)h(of)29888 19649 y Fj(e)29615 19985 y Fq(D)473 b Fs(whic)-36 b(h)437 b(pla)-36 b(y)437 b(a)g(role)g(in)f(the)g(assumptions)0 21535 y(\(and)514 b(pro)36 b(ofs\))515 b(of)h(Prop)36 b(osition)515 b(1.3)h(and)e(Theorem)h(1.4.)823 b(First,)535 b(as)515 b(observ)-36 b(ed)515 b(in)g([DSZ1,)g(DSZ2],)536 b(its)0 23084 y(de\014ning)433 b(equation)15048 25020 y(det)220 b Fr(H)18256 24471 y Fo(c)18719 25020 y Fr(W)181 b Fs(\()p Fr(Z)95 b Fs(\))368 b(=)h(det\()p Fr(H)27368 24471 y Fh(\003)27892 25020 y Fr(H)398 b Fq(\000)295 b(j)p Fr(W)181 b Fq(j)32843 24471 y Fp(2)33369 25020 y Fr(I)104 b Fs(\))369 b(=)f(0)12736 b(\(23\))0 27415 y(is)625 b(real)f(v)-72 b(alued;)720 b(here,)672 b Fr(H)726 b Fs(is)625 b(the)f(holomorphic)g(Hessian)g(\(see)h Fq(x)p Fs(3.2\).)1151 b(Hence,)42477 27079 y Fj(e)42204 27415 y Fq(D)730 b(\032)694 b(I)722 b Fs(is)625 b(a)f(real)0 29153 y(analytic)434 b(h)-36 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Fp(3)26275 27019 y Fs(\()p Fr(X)32 b(;)221 b Fk(C)p Fs(\).)1328 28848 y(By)434 b(the)f(Gri\016ths)g(transv)-36 b(ersalit)g(y)434 b(theorem,)g(see)g([GHJ],)g([CO],)g(\(5.4\)\))g(or)g([W)-108 b(a1)q(,)434 b(W)-108 b(a2],)20224 30802 y Fr(@)72 b Fs(\012)21922 31001 y Fo(z)p 20224 31395 2229 54 v 20384 32612 a Fr(@)g(z)21806 32228 y Fo(j)22955 31701 y Fs(=)368 b Fr(k)25011 31900 y Fo(j)25499 31701 y Fs(\()p Fr(z)59 b Fs(\)\012)28113 31900 y Fo(z)28939 31701 y Fs(+)295 b Fr(\037)31061 31900 y Fo(j)31547 31701 y Fr(;)17780 b Fs(\(35\))0 34449 y(where)353 b Fr(\037)4492 34648 y Fo(j)5347 34449 y Fq(2)369 b Fr(H)7783 33967 y Fp(2)p Fo(;)p Fp(1)7680 34778 y Fo(z)9040 34449 y Fs(\()p Fr(X)104 b Fs(\))353 b(and)f(where)h Fr(k)414 b Fq(2)368 b Fr(C)21083 33967 y Fh(1)22079 34449 y Fs(\()p Fq(M)p Fs(\).)552 b(Note)353 b(that)f(although)36869 33926 y Fo(@)52 b Fp(\012)38097 34037 y Ff(z)p 36869 34144 1705 54 v 36993 34907 a Fo(@)g(z)38016 34655 y Ff(j)39060 34449 y Fs(is)352 b(holomorphic,)370 b(neither)0 35999 y(term)433 b(on)h(the)f(righ)-36 b(t)433 b(hand)g(side)g(is)h(separately)g(holomorphic.)1328 37549 y(W)-108 b(e)457 b(de\014ne)f(a)g(Levi-Civita)j(connection)d(on)h (the)f(bundle)f Fr(H)31398 37067 y Fp(3)p Fo(;)p Fp(0)33063 37549 y Fq(!)409 b(M)457 b Fs(b)-36 b(y)456 b(orthogonally)i(pro)72 b(jecting)0 39099 y(the)465 b(deriv)-72 b(ativ)-36 b(es)8911 38576 y Fo(@)52 b Fp(\012)10139 38687 y Ff(z)p 8911 38793 V 9035 39557 a Fo(@)g(z)10058 39305 y Ff(j)11214 39099 y Fs(on)-36 b(to)465 b Fr(H)15353 38617 y Fp(3)p Fo(;)p Fp(0)16610 39099 y Fs(.)674 b(This)466 b(de\014nes)e(the)h(W)-108 b(eil-P)-36 b(etersson)465 b(connection)g Fq(r)43706 39298 y Fo(W)131 b(P)45972 39099 y Fs(on)466 b Fr(H)48992 38617 y Fp(3)p Fo(;)p Fp(0)50672 39099 y Fq(!)0 40648 y(M)p Fs(,)15108 42244 y Fq(r)16215 42443 y Fo(W)131 b(P)18385 42244 y Fs(:)370 b Fr(C)20142 41696 y Fh(1)21138 42244 y Fs(\()p Fq(M)p Fr(;)221 b Fq(L)p Fs(\))370 b Fq(!)f Fr(C)28336 41696 y Fh(1)29332 42244 y Fs(\()p Fq(M)p Fr(;)221 b Fq(L)296 b(\012)g Fr(T)35499 41696 y Fh(\003)36024 42244 y Fs(\))p Fr(:)0 44111 y Fs(It)434 b(follo)-36 b(ws)435 b(from)f(\(35\))g(that)17489 45523 y Fr(@)p 16914 46116 1910 54 v 16914 47333 a(@)72 b(z)18336 46949 y Fo(j)19177 44613 y Fj(Z)19915 47621 y Fo(X)21035 46421 y Fs(\012)21974 46620 y Fo(z)22800 46421 y Fq(^)p 23981 45354 940 54 v 295 w Fs(\012)24920 46620 y Fo(z)25819 46421 y Fs(=)369 b Fr(k)27876 46620 y Fo(j)28585 44613 y Fj(Z)29323 47621 y Fo(X)30442 46421 y Fs(\012)31381 46620 y Fo(z)32207 46421 y Fq(^)p 33388 45354 V 295 w Fs(\012)34327 46620 y Fo(z)34858 46421 y Fr(;)14469 b Fs(\(36\))0 49045 y(whic)-36 b(h)433 b(b)-36 b(y)434 b(\(34\))g(implies)22733 51355 y Fr(k)23409 51554 y Fo(j)24265 51355 y Fs(=)369 b Fq(\000)26812 50456 y Fr(@)72 b(K)p 26812 51050 1961 54 v 26838 52266 a(@)g(z)28260 51883 y Fo(j)28906 51355 y Fr(:)20421 b Fs(\(37\))0 53602 y(Hence,)14936 55412 y Fq(r)16043 55611 y Fo(W)131 b(P)17844 55412 y Fs(\012)18783 55611 y Fo(z)19683 55412 y Fs(=)368 b Fq(\000)p Fr(@)72 b(K)390 b Fq(\012)296 b Fs(\012)26620 55611 y Fo(z)27520 55412 y Fs(=)28900 54150 y Fj(X)31040 55412 y Fr(k)31716 55611 y Fo(j)32204 55412 y Fr(dz)33484 55611 y Fo(j)34266 55412 y Fq(\012)f Fs(\012)36533 55611 y Fo(z)0 57493 y Fs(is)434 b(the)f(Chern)g(connection)g(of)i(the)e(W)-108 b(eil-P)-36 b(etersson)433 b(Hermitian)h(metric.)1328 59043 y(W)-108 b(e)434 b(also)g(de\014ne)f(the)g(forms)14579 59742 y Fj(8)14579 60938 y(<)14579 63329 y(:)16313 60894 y Fq(D)17338 61093 y Fo(j)17825 60894 y Fs(\012)18764 61093 y Fo(z)19664 60894 y Fs(=)21632 60371 y Fo(@)p 21177 60588 1457 54 v 21177 61352 a(@)52 b(z)22200 61099 y Ff(j)22767 60894 y Fs(\012)295 b(+)25467 60371 y Fo(@)52 b(K)p 25441 60588 V 25441 61352 a(@)g(z)26464 61099 y Ff(j)27030 60894 y Fs(\012)16313 64046 y Fq(D)17338 64245 y Fo(j)17825 64046 y Fq(D)18850 64245 y Fo(k)19419 64046 y Fs(\012)20358 64245 y Fo(z)21257 64046 y Fs(=)369 b(\()23731 63523 y Fo(@)p 23277 63741 V 23277 64504 a(@)52 b(z)24300 64252 y Ff(j)25161 64046 y Fs(+)26627 63523 y Fo(@)g(K)p 26601 63741 V 26601 64504 a(@)g(z)27624 64252 y Ff(j)28190 64046 y Fs(\)\()29826 63523 y Fo(@)p 29335 63741 1532 54 v 29335 64507 a(@)g(z)30358 64255 y Ff(k)31294 64046 y Fs(+)32797 63523 y Fo(@)g(K)p 32734 63741 V 32734 64507 a(@)g(z)33757 64255 y Ff(k)34398 64046 y Fs(\)\012)35843 64245 y Fo(z)36373 64046 y Fr(:)49688 62465 y Fs(\(38\))0 66146 y(W)-108 b(e)434 b(then)e(ha)-36 b(v)g(e)15211 68124 y Fq(D)16236 68323 y Fo(j)16723 68124 y Fs(\012)17662 68323 y Fo(z)18562 68124 y Fs(=)20075 67225 y Fr(@)72 b Fs(\012)21773 67424 y Fo(z)p 20075 67819 2229 54 v 20235 69035 a Fr(@)g(z)21657 68652 y Fo(j)22732 68124 y Fq(\000)295 b Fr(k)24736 68323 y Fo(j)25223 68124 y Fs(\012)26162 68323 y Fo(z)27062 68124 y Fs(=)369 b Fr(\037)29258 68323 y Fo(j)30114 68124 y Fq(2)f Fr(H)32549 67576 y Fp(2)p Fo(;)p Fp(1)33806 68124 y Fs(\()p Fr(X)35391 68323 y Fo(z)35922 68124 y Fs(\))p Fr(:)12899 b Fs(\(39\))0 70681 y(The)413 b(op)36 b(erator)414 b Fq(D)8897 70880 y Fo(j)9383 70681 y Fs(\012)10322 70880 y Fo(z)11266 70681 y Fs(is)g(analogous)g(to)f(the)g(second)g(fundamen)-36 b(tal)412 b(form)i Fr(I)104 b(I)g Fs(\()p Fr(X)32 b(;)221 b(Y)290 b Fs(\))369 b(=)f(\()44639 70345 y(~)44411 70681 y Fq(r)45518 70880 y Fo(X)46416 70681 y Fr(Y)289 b Fs(\))47966 70199 y Fh(?)49166 70681 y Fs(of)414 b(an)0 72231 y(em)-36 b(b)36 b(edding,)407 b(i.e.)568 b(it)401 b(is)h(the)e(`normal')i(comp) 36 b(onen)-36 b(t)400 b(of)i(the)f(am)-36 b(bien)g(t)400 b(deriv)-72 b(ativ)-36 b(e.)568 b(It)401 b(is)h(kno)-36 b(wn)401 b(that)g(the)p eop %%Page: 15 15 15 14 bop 11424 681 a Fm(CRITICAL)440 b(POINTS)j(AND)g(SUPERSYMMETRIC)e (V)-114 b(A)-28 b(CUA,)443 b(I)28 b(I)g(I)10843 b(15)0 3337 y Fs(\014rst)499 b(v)-72 b(ariational)501 b(deriv)-72 b(ativ)-36 b(es)500 b(span)f Fr(H)20109 2855 y Fp(2)p Fo(;)p Fp(1)21866 3337 y Fs(\(see)g(e.g.)777 b([W)-108 b(a1)q(,)500 b(W)-108 b(a2].)777 b(\(In)499 b(the)g(ph)-36 b(ysics)499 b(literature,)516 b Fr(D)51340 3536 y Fo(\013)0 4887 y Fs(is)510 b(often)f(describ)36 b(ed)509 b(as)g(a)h(connection,) 528 b(and)509 b(is)h(often)f(iden)-36 b(ti\014ed)508 b(with)h Fq(r)37611 5086 y Fo(W)131 b(P)39412 4887 y Fs(,)529 b(but)508 b(this)h(is)h(not)f(quite)0 6437 y(correct)434 b(as)f(it)h(is)g(applied)f(to)h(\012)15622 6636 y Fo(z)16153 6437 y Fs(\).)1328 7987 y(The)505 b(W)-108 b(eil-P)-36 b(etersson)504 b(Hermitian)h(metric)23321 6990 y Fj(P)24945 7987 y Fr(G)25971 8241 y Fo(i)26272 8037 y Fp(\026)26291 8241 y Fo(j)26778 7987 y Fr(dz)28058 8186 y Fo(i)28655 7987 y Fr(d)79 b Fs(\026)-729 b Fr(z)29916 8037 y Fp(\026)29935 8241 y Fo(j)30927 7987 y Fs(on)504 b Fq(M)h Fs(is)g(the)f(curv)-72 b(ature)504 b(\(1)p Fr(;)221 b Fs(1\)-form)505 b(of)0 9536 y(the)433 b(Ho)36 b(dge)434 b(bundle.)577 b(F)-108 b(rom)433 b(\(34\))h(and)f(\(39\),)h(w)-36 b(e)434 b(ha)-36 b(v)g(e:)13490 13009 y Fr(G)14516 13293 y Fo(j)14970 13059 y Fp(\026)14948 13293 y Fo(k)15886 13009 y Fs(=)18707 12110 y Fr(@)19466 11628 y Fp(2)p 17399 12703 3901 54 v 17399 13920 a Fr(@)72 b(z)18821 13536 y Fo(j)19308 13920 y Fr(@)151 b Fs(\026)-729 b Fr(z)20730 13536 y Fo(k)21432 13009 y Fr(K)95 b Fs(\()p Fr(z)59 b(;)301 b Fs(\026)-730 b Fr(z)61 b Fs(\))368 b(=)h Fq(\000)28471 10959 y Fj(R)29098 12491 y Fh(M)30495 12029 y Fq(D)31520 12228 y Fo(j)32006 12029 y Fs(\012)32945 12228 y Fo(z)33771 12029 y Fq(^)p 34952 10962 3064 54 v 295 w(D)35977 12228 y Fo(k)36546 12029 y Fs(\012)37485 12228 y Fo(z)p 28471 12703 9545 54 v 30024 12966 a Fj(R)30651 14497 y Fh(M)32047 14036 y Fs(\012)32986 14235 y Fo(z)33812 14036 y Fq(^)p 34993 12969 940 54 v 295 w Fs(\012)35933 14235 y Fo(z)38148 13009 y Fr(:)11179 b Fs(\(40\))0 16974 y(2.3.)652 b FA(Y)-125 b(uk)-83 b(a)-42 b(w)g(a)493 b(couplings)g(and)f(sp)42 b(ecial)492 b(geometry)f(of)g(the)g(mo)42 b(duli)492 b(space.)651 b Fs(In)427 b(Theorem)h(1.5,)0 18523 y(the)421 b(densit)-36 b(y)422 b(of)g(critical)h(p)36 b(oin)-36 b(ts)421 b(is)h(expressed)f(as)h(an)g(in)-36 b(tegral)422 b(o)-36 b(v)g(er)422 b(a)g(space)g Fq(H)40038 18722 y Fo(Z)41064 18523 y Fq(\010)271 b Fk(C)p Fs(,)425 b(where)c Fq(H)48980 18722 y Fo(Z)50157 18523 y Fs(\(cf.)0 20073 y(27\))514 b(is)g(a)f(subspace)g(of)h(the)f(complex)h(symmetric)g (matrices)f(Sym\()p Fr(h)34887 19591 y Fp(2)p Fo(;)p Fp(1)36494 20073 y Fs(+)349 b(1)p Fr(;)221 b Fk(C)p Fs(\))515 b(spanned)d(b)-36 b(y)513 b(certain)0 21731 y(sp)36 b(ecial)555 b(matrices)f Fq(F)10772 21101 y Fo(j)10640 22101 y(ik)11529 21731 y Fs(\()p Fr(z)59 b Fs(\).)940 b(The)554 b(comp)36 b(onen)-36 b(ts)553 b(are)i(kno)-36 b(wn)554 b(as)g(Y)-108 b(uk)-72 b(a)-36 b(w)g(a)554 b(couplings)h(and)e(de\014ned)g(as)0 23281 y(follo)-36 b(ws:)555 b(A)384 b(priori,)394 b Fq(D)10997 23480 y Fo(k)11565 23281 y Fq(D)12590 23480 y Fo(j)13077 23281 y Fs(\012)14016 23480 y Fo(z)14916 23281 y Fq(2)368 b Fr(H)17351 22799 y Fp(2)p Fo(;)p Fp(1)18801 23281 y Fq(\010)193 b Fr(H)21208 22799 y Fp(1)p Fo(;)p Fp(2)22465 23281 y Fs(,)394 b(and)383 b(moreo)-36 b(v)g(er)384 b(its)g Fr(H)34194 22799 y Fp(2)p Fo(;)p Fp(1)35834 23281 y Fs(comp)36 b(onen)-36 b(t)383 b(v)-72 b(anishes)383 b(\(see)h(e.g.)0 24987 y([CO,)434 b(\(5.5\)]\).)579 b(Hence)434 b(w)-36 b(e)433 b(ma)-36 b(y)435 b(de\014ne)d Fq(F)21019 24271 y Fp(\026)21030 24505 y Fo(l)20898 25332 y(k)24 b(j)22332 24987 y Fs(b)-36 b(y)11306 27782 y Fq(D)12331 27981 y Fo(k)12900 27782 y Fq(D)13925 27981 y Fo(j)14411 27782 y Fs(\012)15350 27981 y Fo(z)16250 27782 y Fs(=)369 b Fq(\000)18664 26673 y(p)p 19771 26673 1684 54 v 1109 x(\000)p Fs(1)222 b Fr(e)22279 27233 y Fo(K)23190 27782 y Fq(F)24266 27000 y Fp(\026)24277 27233 y Fo(l)24145 28110 y(k)24 b(j)p 25145 26715 2311 54 v 25145 27782 a Fq(D)26170 27981 y Fo(l)26516 27782 y Fs(\012)2823 b(\(1)370 b Fq(\024)f Fr(j)r(;)221 b(k)45 b(;)221 b(l)400 b Fq(\024)369 b Fr(h)38569 27233 y Fp(2)p Fo(;)p Fp(1)39827 27782 y Fs(\))p Fr(:)8994 b Fs(\(41\))0 30266 y(See)433 b(also)i([St1,)f(\(28\)].)1328 31816 y(The)382 b(Y)-108 b(uk)-72 b(a)-36 b(w)g(a)381 b(couplings)g(are)g(related)h(to)f(the)f (p)36 b(erio)g(ds)382 b(of)f(\012)31363 32015 y Fo(z)32275 31816 y Fs(and)g(to)g(the)g(so-called)g(prep)36 b(oten)-36 b(tial)381 b(of)0 33365 y Fq(M)p Fs(.)559 b(W)-108 b(e)376 b(pause)f(to)g(recall)i(the)d(basic)i(relations)g(and)f(to)h(direct)f (the)g(reader)g(to)g(the)g(relev)-72 b(an)-36 b(t)376 b(references.)1328 34915 y(First,)328 b(w)-36 b(e)301 b(consider)g(p)36 b(erio)g(ds.)534 b(As)301 b(a)g(basis)g(of)h Fr(H)24716 35114 y Fp(3)25242 34915 y Fs(\()p Fr(X)32 b(;)221 b Fk(R)p Fs(\))302 b(w)-36 b(e)301 b(c)-36 b(ho)36 b(ose)301 b(the)g(symplectic)g(basis)h(consisting)0 36465 y(of)618 b(dually)g(paired)f(Lagrangian)h(subspaces)f(of)h Fr(A)p Fs(-cycles)g Fr(A)31031 36664 y Fo(a)32203 36465 y Fs(and)f Fr(B)67 b Fs(-cycles)618 b Fr(B)41305 36664 y Fo(a)41861 36465 y Fs(.)1129 b(The)618 b(p)36 b(erio)g(ds)617 b(of)0 38015 y(\012)939 38214 y Fo(z)1839 38015 y Fq(2)368 b Fr(H)4274 37533 y Fp(3)p Fo(;)p Fp(0)4171 38343 y Fo(z)5531 38015 y Fs(\()p Fr(X)104 b Fs(\))434 b(o)-36 b(v)g(er)434 b(the)f Fr(A)p Fs(-cycles)14229 41142 y Fr(\020)14897 40594 y Fo(a)15821 41142 y Fs(=)17202 39334 y Fj(Z)17940 42341 y Fo(A)18645 42452 y Ff(a)19425 41142 y Fs(\012)20364 41341 y Fo(z)23496 41142 y Fs(\(1)369 b Fq(\024)h Fr(a)e Fq(\024)h Fr(h)29626 40594 y Fp(2)p Fo(;)p Fp(1)31179 41142 y Fs(+)295 b(1)369 b(=)g Fr(b)35439 41341 y Fp(3)35965 41142 y Fr(=)p Fs(2\))0 44362 y(de\014ne)h(holomorphic)h(co)36 b(ordinates)371 b(on)f Fq(L)20579 43880 y Fh(\003)21105 44362 y Fs(.)558 b(Alternately)-108 b(,)384 b(w)-36 b(e)371 b(can)f(view)i(the)e Fr(\020)39045 43880 y Fo(a)39972 44362 y Fs(as)h(`sp)36 b(ecial')372 b(pro)72 b(jectiv)-36 b(e)0 45912 y(co)36 b(ordinates)324 b(on)g Fq(M)p Fs(.)542 b(The)324 b(p)36 b(erio)g(ds)323 b(of)i(\012)20317 46111 y Fo(z)21171 45912 y Fs(o)-36 b(v)g(er)324 b(the)g Fr(B)67 b Fs(-cycles)324 b(are)g(then)f(holomorphic)h(functions)f(of)i(the)0 47461 y Fr(\020)668 46979 y Fo(a)1224 47461 y Fs(.)557 b(The)371 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Fo(b)21351 69372 y Fg(3)21812 69249 y Fo(=)p Fp(2)20891 69751 y Fj(X)20999 72540 y Fo(a)p Fp(=1)23031 69140 y Fj(\022)24009 71013 y Fr(\020)24677 70464 y Fo(a)25532 70677 y Fj(b)25232 71013 y Fr(A)26207 71212 y Fo(a)27058 71013 y Fq(\000)28587 70114 y Fr(@)72 b Fq(F)p 28519 70707 V 28519 71924 a Fr(@)g(\020)29946 71540 y Fo(a)31123 70677 y Fj(b)30856 71013 y Fr(B)31844 71212 y Fo(a)32399 69140 y Fj(\023)33967 71013 y Fr(:)15360 b Fs(\(43\))p eop %%Page: 16 16 16 15 bop 0 681 a Fm(16)6828 b(MICHAEL)340 b(R.)i(DOUGLAS,)g(BERNARD)e (SHIFFMAN,)i(AND)f(STEVE)f(ZELDITCH)0 3337 y Fs(\(See)622 b([CO],)671 b(\(3.8\)\).)1146 b(F)-108 b(urther,)668 b(in)622 b(these)g(co)36 b(ordinates,)671 b(the)622 b(K\177)-650 b(ahler)623 b(p)36 b(oten)-36 b(tial)623 b(\(34\))f(of)i(the)e(W)-108 b(eil-)0 4887 y(P)-36 b(etersson)433 b(metric)h(ma)-36 b(y)434 b(b)36 b(e)433 b(written)g(as)14346 8718 y Fr(K)95 b Fs(\()p Fr(z)59 b(;)301 b Fs(\026)-730 b Fr(z)60 b Fs(\))369 b(=)g Fq(\000)221 b Fs(log)241 b 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b(extra)h(sign)f Fq(\000)9193 54294 y(p)p 10300 54294 V 1043 x(\000)p Fs(1)q(.)565 b(The)393 b(inequalit)-36 b(y)412 b(\(46\))394 b(holds)f(only)h(for)g (primitiv)-36 b(e)394 b(forms,)402 b(but)392 b(in)h(our)g(case)h(all)0 56887 y(harmonic)389 b(3-forms)g(are)g(primitiv)-36 b(e,)399 b(since)389 b(w)-36 b(e)389 b(are)h(assuming)f(that)f Fr(H)34996 56405 y Fp(1)35521 56887 y Fs(\()p Fr(M)66 b(;)221 b Fk(C)p Fs(\))371 b(=)e(0.\))563 b(T)-108 b(o)389 b(restate)g(\(46\))q(:)0 59324 y Ft(Pr)-30 b(oposition)565 b FA(2.1.)582 b Fl(L)-66 b(et)515 b Fs(dim)222 b Fr(X)569 b Fs(=)463 b(3)p Fl(,)529 b(and)517 b(let)f Fr(b)25901 59523 y Fp(1)26426 59324 y Fs(\()p Fr(X)104 b Fs(\))465 b(=)f(0)p Fl(.)752 b(Then)515 b(for)h(e)-66 b(ach)516 b Fr(z)524 b Fq(2)463 b(M)p Fl(,)529 b(the)516 b(Ho)-66 b(dge-)0 60873 y(R)-33 b(iemann)463 b(form)h(is)h(p)-66 b(ositive)463 b(de\014nite)g(on)i Fr(H)22436 60391 y Fp(2)p Fo(;)p Fp(1)22333 61202 y Fo(z)23988 60873 y Fq(\010)296 b Fr(H)26498 60391 y Fp(0)p Fo(;)p 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Fr(d)-72 b(Z)40985 65111 y Fo(j)41767 64912 y Fq(\012)296 b Fr(e)43699 65111 y Fh(L)44770 64912 y Fr(:)4557 b Fs(\(49\))0 68494 y(The)434 b(critical)g(p)36 b(oin)-36 b(t)433 b(equation)h(th)-36 b(us)433 b(reads,)21444 70492 y Fr(@)72 b(f)p 21147 71085 V 21147 72302 a(@)g(Z)22796 72501 y Fo(j)23711 71391 y Fq(\000)296 b Fr(f)26043 70492 y(@)72 b(K)p 25955 71085 V 25955 72302 a(@)g(Z)27604 72501 y Fo(j)28593 71391 y Fs(=)369 b(0)p Fr(:)p eop %%Page: 19 19 19 18 bop 11424 681 a Fm(CRITICAL)440 b(POINTS)j(AND)g(SUPERSYMMETRIC)e (V)-114 b(A)-28 b(CUA,)443 b(I)28 b(I)g(I)10843 b(19)1328 3337 y Fs(The)503 b(Hessian)h(of)f(a)h(holomorphic)f(section)g Fr(s)f Fs(of)i(a)f(general)h(Hermitian)f(holomorphic)g(line)g(bundle)0 4887 y(\()p Fr(L;)221 b(h)p Fs(\))369 b Fq(!)g Fr(M)573 b Fs(at)434 b(a)g(critical)g(p)36 b(oin)-36 b(t)433 b Fr(Z)529 b Fs(is)434 b(the)f(tensor)18823 7164 y Fr(D)36 b Fq(r)p Fr(W)181 b Fs(\()p Fr(Z)95 b Fs(\))369 b Fq(2)g Fr(T)27019 6616 y Fh(\003)27840 7164 y Fq(\012)295 b Fr(T)30111 6616 y Fh(\003)30932 7164 y Fq(\012)g(L)0 9442 y Fs(where)502 b Fr(D)537 b Fs(is)502 b(a)g(connection)g(on)g Fr(T)17361 8960 y Fh(\003)18228 9442 y Fq(\012)342 b(L)p Fs(.)783 b(A)-36 b(t)501 b(a)h(critical)h(p)36 b(oin)-36 b(t)501 b Fr(Z)95 b Fs(,)520 b Fr(D)36 b Fq(r)p Fr(s)p Fs(\()p Fr(Z)95 b Fs(\))501 b(is)h(indep)36 b(enden)-36 b(t)500 b(of)j(the)0 10991 y(c)-36 b(hoice)434 b(of)g(connection)f(on)h Fr(T)14600 10509 y Fh(\003)15126 10991 y Fs(.)578 b(In)434 b(a)f(lo)36 b(cal)435 b(frame)f(and)g(in)f(lo)36 b(cal)435 b(co)36 b(ordinates)434 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)2479 13269 y Fr(D)3595 12720 y Fh(0)3906 13269 y Fq(r)5013 12720 y Fh(0)5323 13269 y Fr(s)p Fs(\()p Fr(Z)7332 13468 y Fp(0)7858 13269 y Fs(\))369 b(=)10113 12007 y Fj(X)10526 14806 y Fo(j)o(;q)12253 13269 y Fr(H)13434 12720 y Fh(0)13331 13597 y Fo(j)51 b(q)14270 13269 y Fr(d)-72 b(Z)15764 13468 y Fo(q)16566 13269 y Fq(\012)295 b Fr(d)-72 b(Z)19388 13468 y Fo(j)20170 13269 y Fq(\012)296 b Fr(e)22102 13468 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27607 y Fr(@)g(K)p 23631 28200 V 23631 29417 a(@)g(Z)25280 29616 y Fo(q)25920 28506 y Fs(\))p Fr(f)142 b Fs(\()p Fr(Z)28605 28705 y Fp(0)29131 28506 y Fs(\))369 b Fr(;)19321 b Fs(\(52\))6266 32238 y Fr(H)7447 31690 y Fh(00)7344 32566 y Fo(j)51 b(q)9390 32238 y Fs(=)1106 b Fq(\000)355 b Fr(f)14714 31339 y(@)15473 30857 y Fp(2)15999 31339 y Fr(K)p 13812 31933 4293 54 v 13812 33178 a(@)72 b(Z)15461 33377 y Fo(j)15948 33178 y Fr(@)16983 32842 y Fs(\026)16707 33178 y Fr(Z)17597 33377 y Fo(q)18237 30312 y Fj(\014)18237 31109 y(\014)18237 31906 y(\014)18237 32703 y(\014)18680 33605 y Fo(Z)19318 33728 y Fg(0)20203 32238 y Fs(=)368 b Fq(\000)p Fr(f)142 b Fs(\()p Fr(Z)24795 32437 y Fp(0)25321 32238 y Fs(\)\002)26839 32437 y Fo(j)51 b(q)27999 32238 y Fr(;)1522 b Fs(\002)30894 32437 y Fo(h)31491 32238 y Fs(\()p Fr(z)32601 32437 y Fp(0)33127 32238 y Fs(\))369 b(=)35383 30976 y Fj(X)35796 33775 y Fo(j)o(;q)37523 32238 y Fs(\002)38535 32437 y Fo(j)51 b(q)39473 32238 y Fr(d)-72 b(Z)40967 32437 y 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45075 y Fh(\003)36722 45886 y Fo(\034)37723 45557 y Fs(is)i(dual)e(to)i(the)e(lo)36 b(cal)427 b(frame)0 47107 y Fr(!)810 47306 y Fo(\034)1754 47107 y Fs(=)369 b Fr(dx)210 b Fs(+)g Fr(\034)148 b(dy)440 b Fs(of)393 b Fr(H)11066 46625 y Fp(1)p Fo(;)p Fp(0)12692 47107 y Fq(!)369 b(E)119 b Fs(.)564 b(W)-108 b(e)392 b(let)h Fr(K)95 b Fs(\()p Fr(Z)g Fs(\))369 b(=)f Fr(K)26225 47306 y Fo(X)27123 47107 y Fs(\()p Fr(z)59 b Fs(\))210 b(+)g Fr(K)31337 47329 y Fo(T)32016 47077 y Fg(2)32532 47107 y Fs(\()p Fr(\034)148 b Fs(\))392 b(b)36 b(e)392 b(the)f(K\177)-650 b(ahler)393 b(p)36 b(oten)-36 b(tial)392 b(for)h(the)0 48657 y(lo)36 b(cal)435 b(frame)f(\012)7659 48856 y Fo(z)8485 48657 y Fq(\012)295 b Fr(!)10623 48856 y Fo(\034)11632 48657 y Fs(of)434 b(the)f(\(p)36 b(ositiv)-36 b(e\))434 b(Ho)36 b(dge)435 b(bundle)d Fq(L)30403 48175 y Fh(\003)30929 48657 y Fs(.)578 b(W)-108 b(e)434 b(then)e(ha)-36 b(v)g(e)11913 51040 y Fq(j)p Fr(e)12885 51239 y Fh(L)13587 51040 y Fs(\()p Fr(Z)95 b Fs(\))p Fq(j)15953 50491 y Fp(2)15953 51368 y Fo(h)16920 51040 y Fs(=)369 b Fq(j)p Fs(\012)19609 51239 y Fo(z)20435 51040 y Fq(\012)295 b Fr(!)22573 51239 y Fo(\034)23148 51040 y Fq(j)23517 50490 y Fh(\000)p Fp(2)23517 51409 y Fo(h)24059 51558 y Ff(W)108 b(P)26037 51040 y Fs(=)368 b Fr(e)28020 50491 y Fo(K)63 b Fp(\()p Fo(Z)g Fp(\))30733 51040 y Fs(=)368 b Fr(e)32716 50491 y Fo(K)33509 50640 y Ff(X)34281 50491 y Fp(\()p Fo(z)37 b Fp(\))35544 51040 y Fr(e)36147 50491 y Fo(K)36940 50805 y Ff(T)37534 50614 y Fg(2)38050 50491 y Fp(\()p Fo(\034)102 b Fp(\))39726 51040 y Fr(:)9601 b Fs(\(54\))1328 53319 y(Hence,)434 b(the)f(W)-108 b(eil-P)-36 b(etersson)433 b(K\177)-650 b(ahler)434 b(p)36 b(oten)-36 b(tial)434 b(on)f(the)g(pro)36 b(duct)433 b(is)14948 56223 y Fr(K)95 b Fs(\()p Fr(Z)g Fs(\))369 b(=)g Fq(\000)221 b Fs(log)23053 54415 y Fj(Z)23791 57423 y Fo(X)24910 56223 y Fs(\012)25849 56422 y Fo(z)26675 56223 y Fq(^)p 27856 55156 940 54 v 295 w Fs(\012)28795 56422 y Fo(z)29621 56223 y Fq(\000)295 b Fs(log)19 b(\()68 b(\026)-718 b Fr(\034)444 b Fq(\000)295 b Fr(\034)148 b Fs(\))p Fr(:)0 59183 y Fs(In)433 b(particular,)h(the)f Fr(\034)148 b Fs(-co)-36 b(v)-72 b(arian)-36 b(t)434 b(deriv)-72 b(ativ)-36 b(e)434 b(on)g Fq(L)g Fs(is)g(giv)-36 b(en)434 b(in)f(the)g(lo)36 b(cal)435 b(frame)f Fr(e)41873 59382 y Fh(L)43009 59183 y Fs(b)-36 b(y)20776 62101 y Fq(r)21883 62300 y Fo(\034)22827 62101 y Fs(=)24697 61202 y Fr(@)p 24340 61795 1473 54 v 24340 63012 a(@)72 b(\034)26240 62101 y Fs(+)28880 61202 y(1)p 27680 61795 3050 54 v 27748 63012 a(\026)-718 b Fr(\034)444 b Fq(\000)295 b Fr(\034)30863 62101 y(:)18464 b Fs(\(55\))0 64883 y(Hence,)434 b(with)f Fr(W)8417 65082 y Fo(G)9575 64883 y Fs(=)368 b Fr(f)11596 65082 y Fo(G)12606 64883 y Fr(e)13209 65082 y Fh(L)13912 64883 y Fs(,)13915 67906 y Fq(r)15022 68105 y Fo(\034)15597 67906 y Fr(f)16238 68105 y Fo(G)18133 67906 y Fs(=)20252 66098 y Fj(Z)20990 69105 y Fo(X)22110 66033 y Fj(\024)22811 67906 y Fr(H)397 b Fs(+)26926 67007 y(1)p 25726 67600 V 25794 68817 a(\026)-718 b Fr(\034)443 b Fq(\000)296 b Fr(\034)28909 67906 y Fs(\()p Fr(F)475 b Fs(+)295 b Fr(\034)148 b(H)103 b Fs(\))34439 66033 y Fj(\025)35435 67906 y Fq(^)294 b Fs(\012)37554 68105 y Fo(z)18133 71474 y Fs(=)21585 70575 y(1)p 20385 71169 V 20453 72385 a(\026)-718 b Fr(\034)443 b Fq(\000)296 b Fr(\034)23789 69666 y Fj(Z)24527 72673 y Fo(X)25425 71474 y Fs(\()p Fr(F)476 b Fs(+)362 b(\026)-717 b Fr(\034)148 b(H)103 b Fs(\))294 b Fq(^)h Fs(\012)33370 71673 y Fo(z)33901 71474 y Fr(:)15426 b Fs(\(56\))p eop %%Page: 20 20 20 19 bop 0 681 a Fm(20)6828 b(MICHAEL)340 b(R.)i(DOUGLAS,)g(BERNARD)e (SHIFFMAN,)i(AND)f(STEVE)f(ZELDITCH)0 3337 y Fs(T)-108 b(o)434 b(compute)f(the)g Fr(z)59 b Fs(-deriv)-72 b(ativ)-36 b(es,)434 b(w)-36 b(e)434 b(see)g(from)g Fq(x)p Fs(2.2)h(and)e (\(48\){\(49\))i(that)6026 6419 y Fq(r)7133 6653 y Fo(z)7608 6401 y Ff(j)8097 6419 y Fr(f)8738 6618 y Fo(G)10634 6419 y Fs(=)12752 4546 y Fj(\022)13863 5520 y Fr(@)72 b(f)15263 5719 y Fo(G)p 13863 6113 2189 54 v 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Fj(\003)31262 49678 y Fk(r)295 b Fq(f)p Fs(0)p Fq(g)0 51836 y Fs(is)434 b(op)36 b(en.)578 b(By)434 b(Prop)36 b(osition)434 b(3.2,)h(it)f(su\016ces)f(to)h(sho)-36 b(w)433 b(that)g(the)g(image)i(of)f(the)f(map)3558 53994 y Fr(\023)369 b Fs(:)g Fr(U)434 b Fq(\002)296 b Fs([)p Fr(H)9305 53445 y Fp(3)9830 53994 y Fs(\()p Fr(X)32 b(;)221 b Fk(R)p Fs(\))297 b Fk(r)e Fq(f)p Fs(0)p Fq(g)p Fs(])370 b Fq(!)f Fr(H)20706 53445 y Fp(3)21232 53994 y Fs(\()p Fr(X)32 b(;)221 b Fk(C)p Fs(\))p Fr(;)960 b(\023)p Fs(\()p Fr(z)59 b(;)221 b(\034)76 b(;)221 b(H)103 b Fs(\))370 b(=)e Fr(\023)33548 54193 y Fo(z)37 b(;\034)34860 53994 y Fs(\()p Fr(H)103 b Fs(\))368 b(=)g Fr(F)181 b Fs(\()p Fr(z)59 b(;)221 b(\034)76 b(;)221 b(H)103 b Fs(\))296 b(+)e Fr(iH)471 b(;)0 56151 y Fs(is)571 b(op)36 b(en.)988 b(W)-108 b(e)571 b(\014x)f(\()p Fr(z)10970 56350 y Fp(0)11496 56151 y Fr(;)221 b(\034)12643 56350 y Fp(0)13170 56151 y Fr(;)g(H)14830 56350 y Fp(0)15356 56151 y Fs(\))570 b(and)g(consider)g(the)g(deriv)-72 b(ativ)-36 b(e)571 b Fr(D)36 b(\023)p Fq(j)34720 56350 y Fo(z)35158 56473 y Fg(0)35619 56350 y Fo(;\034)36298 56473 y Fg(0)36759 56350 y Fo(;H)37793 56473 y Fg(0)38880 56151 y Fs(on)571 b Fr(T)41586 56350 y Fo(z)42024 56473 y Fg(0)42485 56350 y Fo(;\034)43164 56473 y Fg(0)43680 56151 y Fq(C)467 b(\002)388 b Fr(H)47448 55669 y Fp(3)47973 56151 y Fs(\()p Fr(X)32 b(;)221 b Fk(R)p Fs(\).)0 57701 y(since)413 b(the)f(linear)h (map)g Fr(\023)12307 57900 y Fo(z)37 b(;\034)14031 57701 y Fs(is)413 b(bijectiv)-36 b(e,)418 b(if)413 b(w)-36 b(e)413 b(v)-72 b(ary)414 b Fr(H)103 b Fs(,)416 b(w)-36 b(e)413 b(get)g(all)h(of)g Fr(H)37191 57219 y Fp(3)37088 58030 y Fo(z)37 b(;\034)38399 57701 y Fs(,)417 b(so)c(the)f(issue)h(is) g(to)g(pro)-36 b(v)g(e)0 59251 y(that)433 b(w)-36 b(e)434 b(obtain)f(the)h(complemen)-36 b(tary)433 b(space)h(b)-36 b(y)433 b(taking)i(v)-72 b(ariations)434 b(in)g Fr(\034)76 b(;)221 b(z)59 b Fs(.)1328 60801 y(First,)434 b Fr(H)6039 60319 y Fp(3)5936 61129 y Fo(z)37 b(;\034)7617 60801 y Fs(=)368 b Fr(I)9675 60319 y Fh(\000)p Fp(1)9571 61129 y Fo(\034)10933 60801 y Fs(\()p Fr(H)12620 60319 y Fp(2)p Fo(;)p Fp(1)12517 61129 y Fo(z)14172 60801 y Fq(\010)295 b Fr(H)16681 60319 y Fp(0)p Fo(;)p Fp(3)16578 61129 y Fo(z)17938 60801 y Fs(\))p Fr(:)434 b Fs(The)g Fr(z)493 b Fs(v)-72 b(ariations)435 b(of)f Fr(H)31641 60319 y Fp(2)p Fo(;)p Fp(1)31538 61129 y Fo(z)33193 60801 y Fq(\010)296 b Fr(H)35703 60319 y Fp(0)p Fo(;)p Fp(3)35600 61129 y Fo(z)37393 60801 y Fs(span)433 b(this)h(space)g(plus)f Fr(H)50382 60319 y Fp(1)p Fo(;)p Fp(2)50279 61129 y Fo(z)51639 60801 y Fs(.)0 62465 y(By)359 b(\(61\){\(63\))q(,)g(v)-72 b(ariations)341 b(in)f(Re)222 b Fr(\034)148 b Fs(,)359 b(resp.)547 b(Im)222 b Fr(\034)148 b Fs(,)359 b(pro)36 b(duce)339 b(Re)222 b(\012)32438 62664 y Fo(z)32969 62465 y Fr(;)f Fs(Im)h(\012)36265 62664 y Fo(z)37136 62465 y Fs(and)340 b(hence)f Fr(H)44272 61983 y Fp(3)p Fo(;)p Fp(0)44169 62793 y Fo(z)45898 62465 y Fs(=span\(\012)50964 62664 y Fo(z)51494 62465 y Fs(\))0 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Fq(S)14949 3536 y Fo(Z)15705 3337 y FA(.)650 b Fs(In)363 b(Theorem)f(1.4,)378 b(w)-36 b(e)363 b(ha)-36 b(v)g(e)363 b(expressed)f Fq(N)37767 3536 y Fo( )38469 3337 y Fs(\()p Fr(L)p Fs(\))g(in)h(terms)f(of)h(a)g (Gauss-)0 4887 y(ian)309 b(t)-36 b(yp)36 b(e)309 b(ensem)-36 b(ble)308 b(of)i(holomorphic)f(sections)g(in)f Fq(S)25857 5086 y Fo(Z)26613 4887 y Fs(.)537 b(W)-108 b(e)309 b(no)-36 b(w)309 b(sp)36 b(ecify)310 b(the)e(inner)g(pro)36 b(duct,)333 b(Gaussian)0 6437 y(measure)433 b(and)g(Szeg\177)-650 b(o)434 b(k)-36 b(ernel)434 b(on)g(this)f(space.)0 8954 y Ft(Pr)-30 b(oposition)478 b FA(3.4.)540 b Fl(The)447 b(Ho)-66 b(dge-R)-33 b(iemann)444 b(Hermitian)g(inner)i(pr)-66 b(o)g(duct)447 b(on)g Fr(H)40755 8472 y Fp(3)41280 8954 y Fs(\()p Fr(X)32 b(;)221 b Fk(C)p Fs(\))448 b Fl(r)-66 b(estricts)446 b(for)0 10504 y(e)-66 b(ach)507 b Fr(Z)543 b Fs(=)448 b(\()p Fr(z)59 b(;)221 b(\034)148 b Fs(\))508 b Fl(to)f(de\014ne)g(a)g(c)-66 b(omplex)507 b(value)-66 b(d)507 b(inner)f(pr)-66 b(o)g(duct)507 b(on)g Fr(H)36129 10022 y Fp(3)36026 10840 y Fo(Z)37289 10504 y Fl(which)g(satis\014es)g Fr(Q)46938 10703 y Fo(Z)47694 10504 y Fs([)p Fr(G)p Fs(])448 b Fr(>)g Fs(0)0 12054 y Fl(for)465 b(al)66 b(l)465 b Fr(G)369 b Fq(6)p Fs(=)g(0)p Fl(.)597 b(Mor)-66 b(e)g(over,)463 b(the)i(map)g Fr(I)20059 12253 y Fo(\034)21003 12054 y Fs(:)369 b Fr(H)22914 11572 y Fp(3)22811 12389 y Fo(Z)23935 12054 y Fq(!)h Fr(H)26814 11572 y Fp(2)p Fo(;)p Fp(1)26711 12382 y Fo(z)28366 12054 y Fq(\010)295 b Fr(H)30875 11572 y Fp(0)p Fo(;)p Fp(3)30772 12382 y Fo(z)32597 12054 y Fl(satis\014es)465 b Fr(Q)p Fs([)p Fr(I)39444 12253 y Fo(\034)40019 12054 y Fr(G)p Fs(])369 b(=)g(Im)221 b Fr(\034)517 b(Q)p Fs([)p Fr(G)p Fs(])p Fr(:)0 14571 y Fl(Pr)-66 b(o)g(of.)552 b Fs(It)434 b(follo)-36 b(ws)435 b(b)-36 b(y)434 b(Prop)36 b(osition)434 b(2.1)g(that)g(the)f(symmetric) g(bilinear)h(form)9005 17627 y Fr(Q)p Fs([)p Fr(F)476 b Fs(+)295 b Fr(\034)148 b(H)103 b Fs(])368 b(=)h Fr(i)17469 17079 y Fp(3)18216 15819 y Fj(Z)18954 18827 y Fo(X)19852 17627 y Fs(\()p Fr(F)476 b Fs(+)295 b Fr(\034)148 b(H)103 b Fs(\))294 b Fq(^)p 26858 16472 5530 54 v 295 w Fs(\()p Fr(F)476 b Fs(+)294 b Fr(\034)148 b(H)103 b Fs(\))369 b(=)f(Im)222 b Fr(\034)517 b(Q)p Fs([)p Fr(F)476 b Fs(+)295 b Fr(iH)103 b Fs(])6692 b(\(64\))0 20869 y(on)433 b Fr(H)2987 20387 y Fp(3)2884 21198 y Fo(z)37 b(;\034)4196 20869 y Fs(\()p Fr(X)32 b(;)221 b Fk(C)p Fs(\))435 b(in)e(\(59\))h(is)g(p)36 b(ositiv)-36 b(e)434 b(de\014nite.)27736 b Fc(\003)1328 23496 y Fs(Recall)435 b(that)e(w)-36 b(e)434 b(ha)-36 b(v)g(e)434 b(the)f(real-linear)h(isomorphisms)17134 26235 y Fr(H)18315 25753 y Fp(3)18840 26235 y Fs(\()p Fr(X)32 b(;)221 b Fk(C)p Fs(\))24178 25482 y Fh(W)23612 26235 y Fq(\000)-221 b(!)803 b(S)469 b(\032)369 b Fr(H)30411 25687 y Fp(0)30936 26235 y Fs(\()p Fq(C)77 b Fr(;)221 b Fq(L)p Fs(\))18839 28173 y Fr(I)19413 28372 y Fo(\034)20135 28173 y Fq(#)14813 b Fr(:)13715 b Fs(\(65\))17134 30110 y Fr(H)18315 29628 y Fp(3)18840 30110 y Fs(\()p Fr(X)32 b(;)221 b Fk(C)p Fs(\))0 32539 y(where)456 b Fr(I)4354 32738 y Fo(\034)4929 32539 y Fs(\()p Fr(F)491 b Fs(+)310 b Fr(iH)103 b Fs(\))406 b(=)h Fr(F)491 b Fs(+)311 b Fr(\034)148 b(H)103 b Fs(.)645 b(Restricting)475 b(\(65\))456 b(to)g(\015uxes)g (with)g(a)h(critical)g(p)36 b(oin)-36 b(t)456 b(at)g Fr(Z)503 b Fs(=)407 b(\()p Fr(z)59 b(;)221 b(\034)148 b Fs(\),)0 34089 y(w)-36 b(e)434 b(ha)-36 b(v)g(e)434 b(isomorphisms)22903 36828 y Fr(H)24084 36346 y Fp(3)23981 37164 y Fo(Z)26342 36075 y Fh(W)25776 36828 y Fq(\000)-221 b(!)803 b(S)29523 37027 y Fo(Z)22189 38765 y Fr(I)22763 38964 y Fo(\034)23485 38765 y Fq(#)8027 b Fr(:)17151 b Fs(\(66\))20570 40702 y Fr(H)21751 40220 y Fp(2)p Fo(;)p Fp(1)21648 41031 y Fo(z)23303 40702 y Fq(\010)295 b Fr(H)25812 40220 y Fp(0)p Fo(;)p Fp(3)25709 41031 y Fo(z)1328 43316 y Fs(W)-108 b(e)384 b(let)5602 42980 y Fj(e)5347 43316 y Fr(Q)h Fs(denote)e(the)g(Hermitian)h(inner)g(pro)36 b(duct)383 b(on)h Fr(H)30262 42834 y Fp(2)p Fo(;)p Fp(1)30159 43644 y Fo(z)31712 43316 y Fq(\010)194 b Fr(H)34120 42834 y Fp(0)p Fo(;)p Fp(3)34017 43644 y Fo(z)35761 43316 y Fs(transp)36 b(orted)383 b(from)i(\()p Fr(H)47473 42834 y Fp(3)47370 43652 y Fo(Z)48125 43316 y Fr(;)221 b(Q)p Fs(\))384 b(b)-36 b(y)0 44866 y Fr(I)574 45065 y Fo(\034)1149 44866 y Fs(;)434 b(i.e.,)14648 46531 y Fj(e)14394 46867 y Fr(Q)p Fs([)p Fr(C)95 b Fs(])369 b(=)g Fr(Q)20173 45791 y Fj(\002)20727 46867 y Fr(I)21405 46318 y Fh(\000)p Fp(1)21301 47195 y Fo(\034)22663 46867 y Fs(\()p Fr(C)95 b Fs(\))24701 45791 y Fj(\003)25844 46867 y Fr(;)1522 b(C)464 b Fq(2)369 b Fr(H)31558 46318 y Fp(2)p Fo(;)p Fp(1)31455 47195 y Fo(z)33110 46867 y Fq(\010)295 b Fr(H)35619 46318 y Fp(0)p Fo(;)p Fp(3)35516 47195 y Fo(z)37245 46867 y Fr(:)12082 b Fs(\(67\))0 49047 y(Hence)433 b(b)-36 b(y)452 b(\(64\),)434 b(w)-36 b(e)434 b(ha)-36 b(v)g(e:)20121 51122 y Fr(Q)p Fs([)p Fr(C)95 b Fs(])370 b(=)e(\(Im)222 b Fr(\034)148 b Fs(\))28624 50786 y Fj(e)28370 51122 y Fr(Q)p Fs([)p Fr(C)95 b Fs(])370 b Fr(:)17809 b Fs(\(68\))0 54121 y(3.6.)1795 b FA(Example)658 b(with)h Fr(h)14456 53639 y Fp(21)16058 54121 y Fs(=)605 b(0)p FA(.)651 b Fs(The)572 b(simplest)h(example)g(is)g(the)f(case)h(where)g(the)f (Calabi-Y)-108 b(au)0 55670 y(manifold)371 b Fr(X)474 b Fs(is)370 b(rigid,)383 b(i.e.)558 b Fq(M)369 b Fs(=)g Fq(f)p Fr(pt)p Fq(g)h Fs(\(see)f([AD)q(,)h(DD1)q(])g(for)h(further)d (details)j(and)e(computer)g(graphics)0 57220 y(of)400 b(critical)h(p)36 b(oin)-36 b(ts)399 b(in)h(this)f(case\).)568 b(Then)399 b(only)h(the)g(parameter)f Fr(\034)517 b Fq(2)369 b(H)412 b Fs(v)-72 b(aries.)568 b(Let)399 b Fr(G)369 b Fs(=)g Fr(F)406 b Fs(+)226 b Fr(iH)103 b Fs(,)405 b(and)0 58770 y(consider)433 b(the)g(\015ux)h(sup)36 b(erp)g(oten)-36 b(tial)433 b Fr(W)19669 58969 y Fo(G)20457 58770 y Fs(.)578 b(Its)434 b(critical)h(p)36 b(oin)-36 b(t)433 b(equation)h(is)21710 61199 y Fr(F)476 b Fs(+)295 b Fr(\034)148 b(H)471 b Fq(2)369 b Fr(H)29033 60651 y Fp(0)p Fo(;)p Fp(3)0 63628 y Fs(\(since)433 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b Fs(+)294 b Fr(\034)148 b(H)472 b Fq(2)368 b Fr(H)24370 71214 y Fp(0)p Fo(;)p Fp(3)26365 71762 y Fq(\()-221 b(\))739 b Fr(A)295 b Fs(+)g Fr(\034)148 b(B)436 b Fs(=)369 b(0)738 b Fq(\()-221 b(\))738 b Fr(\034)517 b Fs(=)369 b Fq(\000)43864 70863 y Fr(A)p 43824 71457 1056 54 v 43824 72673 a(B)45012 71762 y(:)p eop %%Page: 23 23 23 22 bop 11424 681 a Fm(CRITICAL)440 b(POINTS)j(AND)g(SUPERSYMMETRIC)e (V)-114 b(A)-28 b(CUA,)443 b(I)28 b(I)g(I)10843 b(23)1328 3337 y Fs(Eac)-36 b(h)559 b(\015ux)g(sup)36 b(erp)g(oten)-36 b(tial)558 b Fr(W)17301 3536 y Fo(A;B)19659 3337 y Fq(2)582 b(S)659 b Fs(\(with)559 b Fr(A;)221 b(B)650 b Fq(2)582 b Fk(C)p Fs(\))559 b(has)g(a)h(unique)e(critical)i(p)36 b(oin)-36 b(t)559 b(in)g Fq(H)13 b Fs(,)0 4887 y(whic)-36 b(h)433 b(ma)-36 b(y)434 b(or)g(ma)-36 b(y)434 b(not)g(lie)g(in)f(the)g (fundamen)-36 b(tal)433 b(domain)h Fq(C)77 b Fs(.)579 b(In)433 b(the)g(notation)h(of)g(\(15\),)18682 7872 y Fr(\031)48 b Fs(\()p Fq(S)100 b Fs(\))368 b(=)h Fq(f)p Fr(W)25025 8071 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y Fo(L)23767 40026 y Fj(X)23709 42815 y Fo(m)p Fp(=1)25965 41288 y Fr(\033)48 b Fs(\()p Fr(m)p Fs(\))368 b Fq(\030)30805 40389 y Fr(\031)31591 39907 y Fp(2)p 30805 40982 1312 54 v 30810 42199 a Fs(12)32249 41288 y Fr(L)33134 40739 y Fp(2)33955 41288 y Fs(+)295 b Fr(O)36 b Fs(\()p Fr(L)221 b Fs(log)240 b Fr(L)p Fs(\))p Fr(:)8131 b Fs(\(69\))0 45101 y(As)470 b(v)-36 b(eri\014ed)469 b(in)h([DD1)q(])g(\(and)g(as)g (follo)-36 b(ws)471 b(v)-36 b(ery)471 b(simply)f(from)g(Theorem)g (1.4\),)480 b(the)470 b(critical)g(p)36 b(oin)-36 b(ts)470 b(are)0 46651 y(uniformly)434 b(distributed)e(relativ)-36 b(e)435 b(to)f(the)f(h)-36 b(yp)36 b(erb)g(olic)433 b(area)i(form.)7664 49838 y(4.)651 b Ft(Counting)501 b(critical)i(points:)658 b(pr)-30 b(oof)503 b(of)f(Pr)-30 b(oposition)504 b(1.3)1328 52162 y Fs(W)-108 b(e)392 b(no)-36 b(w)392 b(pro)-36 b(v)g(e)391 b(the)g(\014rst)g(result)g(on)g(coun)-36 b(ting)391 b(critical)i(p)36 b(oin)-36 b(ts)391 b(of)h(\015ux)f(sup)36 b(erp)g(oten)-36 b(tials)391 b Fr(W)47496 52361 y Fo(G)48676 52162 y Fs(where)0 53712 y Fr(G)450 b Fs(satis\014es)g(the)g(tadp)36 b(ole)450 b(constrain)-36 b(t)449 b(\(1\).)628 b(Before)450 b(starting)g(the)g(pro)36 b(of,)455 b(w)-36 b(e)450 b(review)h(the)e (geometry)i(of)0 55262 y(the)433 b(lattice)h(p)36 b(oin)-36 b(t)433 b(problem)h(and)f(the)g(critical)h(p)36 b(oin)-36 b(t)433 b(problem.)1328 56811 y(W)-108 b(e)438 b(wish)f(to)g(coun)-36 b(t)437 b(v)-72 b(acua)438 b(in)f(a)g(region)h(of)g(mo)36 b(duli)437 b(space)g(as)h Fr(G)f Fs(v)-72 b(aries)438 b(o)-36 b(v)g(er)438 b(\015uxes)f(satisfying)i(the)0 58361 y(tadp)36 b(ole)513 b(constrain)-36 b(t.)817 b(Equiv)-72 b(alen)-36 b(tly)-108 b(,)534 b(w)-36 b(e)513 b(coun)-36 b(t)513 b(inequiv)-72 b(alen)-36 b(t)513 b(v)-72 b(acua)514 b(in)f(T)-108 b(eic)-36 b(hm)36 b(\177)-686 b(uller)513 b(space.)817 b(That)0 59911 y(is,)434 b(\000)f(acts)h(on)g(the)f(pairs) g(\()p Fr(W)-36 b(;)221 b(Z)95 b Fs(\))435 b(of)f(sup)36 b(erp)g(oten)-36 b(tials)433 b(and)g(mo)36 b(duli)434 b(b)-36 b(y)17847 62196 y Fr(\015)367 b Fq(\001)296 b Fs(\()p Fr(G;)221 b(Z)95 b Fs(\))370 b(=)e(\()p Fr(')p Fs(\()p Fr(\015)72 b Fs(\))295 b Fq(\001)g Fr(G;)221 b(\015)368 b Fq(\001)295 b Fr(Z)95 b Fs(\))p Fr(;)0 64482 y Fs(Therefore)349 b(\000)f(acts)g(on)h(the)e(incidence)h(relation)h (\(14\).)551 b(W)-108 b(e)348 b(only)h(wish)g(to)f(coun)-36 b(t)348 b(critical)h(p)36 b(oin)-36 b(ts)348 b(mo)36 b(dulo)0 66032 y(the)480 b(action)g(of)h(\000.)719 b(T)-108 b(o)480 b(do)g(this,)492 b(there)480 b(are)g(t)-36 b(w)g(o)481 b(c)-36 b(hoices:)672 b(w)-36 b(e)480 b(could)h(break)f(the)g(symmetry) h(b)-36 b(y)480 b(\014xing)0 67581 y(a)492 b(fundamen)-36 b(tal)492 b(domain)g Fq(D)14424 67780 y Fp(\000)15535 67581 y Fq(\032)468 b(C)570 b Fs(for)493 b(\000)e(in)h Fq(C)77 b Fs(,)508 b(i.e.)754 b(only)493 b(coun)-36 b(t)491 b(critical)i(p)36 b(oin)-36 b(ts)491 b(in)h(a)h(fundamen)-36 b(tal)0 69131 y(domain.)620 b(Or)447 b(w)-36 b(e)448 b(could)f(\014x)g(a)h(fundamen)-36 b(tal)447 b(domain)g(for)h Fr(')p Fs(\(\000\))f(in)g Fr(H)35506 68649 y Fp(3)36031 69131 y Fs(\()p Fr(X)32 b(;)221 b Fk(C)p Fs(\))449 b(and)e(coun)-36 b(t)447 b(all)h(critical)0 70681 y(p)36 b(oin)-36 b(ts)395 b(of)i(these)e(sp)36 b(ecial)396 b(\015ux)g(sup)36 b(erp)g(oten)-36 b(tials.)565 b(When)395 b(w)-36 b(e)396 b(do)f(not)h(kno)-36 b(w)396 b(the)f(group)g Fr(')p Fs(\(\000\))g(precisely)-108 b(,)0 72231 y(it)594 b(seems)h(simpler)f(to)g(tak)-36 b(e)595 b(the)e(\014rst)h(option)g(and)g(that)f(is)i(what)f(w)-36 b(e)595 b(do)f(in)g(Prop)36 b(osition)594 b(1.3)h(and)p eop %%Page: 24 24 24 23 bop 0 681 a Fm(24)6828 b(MICHAEL)340 b(R.)i(DOUGLAS,)g(BERNARD)e (SHIFFMAN,)i(AND)f(STEVE)f(ZELDITCH)0 3337 y Fs(Theorem)416 b(1.4.)573 b(W)-108 b(e)416 b(note)f(that)g(the)g(n)-36 b(um)g(b)36 b(er)414 b(of)i(critical)h(p)36 b(oin)-36 b(ts)415 b(of)i Fr(W)35264 3536 y Fo(G)36468 3337 y Fs(in)e(T)-108 b(eic)-36 b(hm)36 b(\177)-686 b(uller)416 b(space)f(equals)0 4887 y(the)433 b(n)-36 b(um)g(b)36 b(er)432 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Fr(\037)13524 21387 y Fo(K)14900 21188 y Fl(wher)-66 b(e)465 b Fr(K)f Fq(\032)369 b(I)563 b Fl(is)465 b(as)g(in)f(Pr)-66 b(op)g(osition)464 b(1.3.)597 b(Then)11375 24337 y Fq(N)12465 24536 y Fo( )13167 24337 y Fs(\()p Fr(L)p Fs(\))369 b(=)f Fr(L)17698 23789 y Fo(b)18101 23912 y Fg(3)18839 22464 y Fj(\024)19540 22529 y(Z)20278 25537 y Fh(S)20968 24337 y Fq(h)p Fr(C)22416 24536 y Fo(W)23489 24337 y Fr(;)221 b( )48 b Fq(i)p Fr(\037)26297 24536 y Fo(Q)27091 24337 y Fs(\()p Fr(W)181 b Fs(\))221 b Fr(dW)476 b Fs(+)294 b Fr(O)34671 23262 y Fj(\000)35280 24337 y Fr(L)36165 23789 y Fh(\000)p Fp(1)p Fo(=)p Fp(2)38363 23262 y Fj(\001)38972 22464 y(\025)40264 24337 y Fr(:)0 27586 y Fl(Pr)-66 b(o)g(of.)552 b Fs(W)-108 b(e)434 b(consider)f(the)g (in)-36 b(teger-v)-72 b(alued)433 b(function)6832 30204 y Fr(f)142 b Fs(\()p Fr(W)181 b Fs(\))368 b(=)h Fq(h)p Fr(C)13233 30403 y Fo(W)14306 30204 y Fr(;)221 b( )48 b Fq(i)368 b Fs(=)20258 28942 y Fj(X)18048 31821 y Fh(f)p Fo(Z)63 b Fp(:)p Fh(r)p Fo(W)131 b Fp(\()p Fo(Z)63 b Fp(\)=0)p Fh(g)24608 30204 y Fr( )48 b Fs(\()p Fr(Z)23 b(;)221 b(W)181 b Fs(\))369 b(=)f(#)p Fq(f)p Fr(Z)465 b Fq(2)369 b(C)447 b Fs(:)369 b(\()p Fr(Z)23 b(;)221 b(W)181 b Fs(\))369 b Fq(2)g Fr(K)95 b Fq(g)p Fr(:)0 34371 y Fs(W)-108 b(e)480 b(note)h(that)f(the)g(c)-36 b(haracteristic)480 b(function)g(of)i(the)e(set)g Fq(f)p Fs(0)449 b Fq(\024)g Fr(Q)p Fs([)p Fr(W)181 b Fs(])449 b Fq(\024)g Fr(L)p Fq(g)481 b Fs(is)g Fr(\037)41985 34570 y Fo(Q)42780 34371 y Fs(\()p Fr(W)36 b(=)45200 33246 y Fq(p)p 46307 33246 885 54 v 1125 x Fr(L)p Fs(\).)719 b(Using)0 36003 y(our)433 b(symplectic)h(basis)g(to)g(iden)-36 b(tify)434 b Fr(H)19412 35521 y Fp(3)19937 36003 y Fs(\()p Fr(X)32 b(;)221 b Fk(Z)296 b Fq(\010)24646 34959 y(p)p 25753 34959 1684 54 v 1044 x(\000)p Fs(1)q Fk(Z)p Fs(\))433 b(with)g Fk(Z)33110 35521 y Fp(2)p Fo(b)33983 35644 y Fg(3)34499 36003 y Fs(,)h(w)-36 b(e)434 b(ha)-36 b(v)g(e)2541 38695 y Fq(N)3631 38894 y Fo( )4333 38695 y Fs(\()p Fr(L)p Fs(\))368 b(=)8692 37433 y Fj(X)7979 40403 y Fo(N)94 b Fh(2)p Fi(Z)10037 40108 y Fg(2)p Ff(b)10807 40307 y Fg(3)11545 38695 y Fr(f)142 b Fs(\()p Fr(N)d Fs(\))p Fr(\037)15336 38894 y Fo(Q)16130 38695 y Fs(\()p Fr(N)30 b(=)18358 37506 y Fq(p)p 19465 37506 885 54 v 1189 x Fr(L)p Fs(\))369 b(=)23319 37433 y Fj(X)22606 40403 y Fo(N)94 b Fh(2)p Fi(Z)24664 40108 y Fg(2)p Ff(b)25434 40307 y Fg(3)26172 38695 y Fr(f)142 b Fs(\()p Fr(N)30 b(=)29183 37506 y Fq(p)p 30290 37506 V 1189 x Fr(L)p Fs(\))p Fr(\037)32496 38894 y Fo(Q)33291 38695 y Fs(\()p Fr(N)g(=)35519 37506 y Fq(p)p 36626 37506 V 1189 x Fr(L)p Fs(\))369 b(=)40480 37433 y Fj(X)39766 40403 y Fo(N)94 b Fh(2)p Fi(Z)41824 40108 y Fg(2)p Ff(b)42594 40307 y Fg(3)43333 38695 y Fr(g)48 b Fs(\()p Fr(N)30 b(=)46232 37506 y Fq(p)p 47338 37506 V 47338 38695 a Fr(L)p Fs(\))369 b Fr(;)0 42480 y Fs(where)23302 44342 y Fr(g)416 b Fs(=)369 b Fr(f)142 b(\037)27320 44541 y Fo(Q)28337 44342 y Fr(:)1328 46475 y Fs(W)-108 b(e)453 b(note)f(that)g Fr(f)594 b Fs(is)453 b(constan)-36 b(t)452 b(on)g(eac)-36 b(h)452 b(connected)g(comp)36 b(onen)-36 b(t)452 b(of)h Fq(S)408 b Fk(r)308 b Fs([)p Fq(D)345 b([)307 b Fr(\031)48 b Fs(\()p Fr(@)72 b(K)95 b Fs(\)].)635 b(Since)452 b(the)0 48025 y(n)-36 b(um)g(b)36 b(er)327 b(of)j(comp)36 b(onen)-36 b(ts)328 b(of)i Fq(S)181 b Fk(r)82 b Fq(D)366 b Fs(is)329 b(\014nite,)350 b Fr(f)471 b Fs(is)329 b(b)36 b(ounded.)542 b(W)-108 b(e)329 b(let)g Fr(S)77 b Fs(\()p Fq(S)36927 48224 y Fo(Z)37683 48025 y Fs(\))369 b(=)g Fq(f)p Fr(N)507 b Fq(2)369 b(S)44211 48224 y Fo(Z)45336 48025 y Fs(:)g Fq(k)p Fr(N)139 b Fq(k)369 b Fs(=)f(1)p Fq(g)p Fs(,)0 49574 y(where)576 b Fq(k)p Fr(N)139 b Fq(k)575 b Fs(denotes)h(the)f (norm)h(in)g Fk(Z)20298 49092 y Fp(2)p Fo(b)21171 49215 y Fg(3)21687 49574 y Fs(.)1005 b(Since)576 b Fr(Q)27622 49773 y Fo(Z)28953 49574 y Fs(is)h(p)36 b(ositiv)-36 b(e)577 b(de\014nite,)610 b(the)576 b(sphere)f Fr(S)77 b Fs(\()p Fq(S)49288 49773 y Fo(Z)50044 49574 y Fs(\))576 b(is)0 51124 y(con)-36 b(tained)433 b(in)h(the)f(in)-36 b(terior)433 b(of)h(the)f(ligh)-36 b(t)434 b(cone)g Fq(f)p Fr(W)550 b Fq(2)368 b(S)469 b Fs(:)369 b Fr(Q)p Fs([)p Fr(W)181 b Fs(])370 b Fq(\025)f Fs(0)p Fq(g)p Fs(.)579 b(Let)17425 53542 y Fr(A)18400 53741 y Fo( )19471 53542 y Fs(=)2107 b(sup)20851 54686 y Fo(Z)63 b Fh(2)p Fo(\032)p Fp(\(Supp)155 b Fo( )32 b Fp(\))26508 53542 y Fq(k)p Fr(Q)28202 52992 y Fh(\000)p Fp(1)28202 53902 y Fo(Z)29460 53542 y Fq(k)370 b Fr(<)e Fs(+)p Fq(1)p Fr(:)15113 b Fs(\(70\))0 56928 y(Then)12014 60074 y(inf)13809 57351 y Fj(8)13809 58547 y(<)13809 60938 y(:)14990 60074 y Fr(Q)p Fs([)p Fr(W)181 b Fs(])370 b(:)f Fr(W)549 b Fq(2)24263 58812 y Fj([)22283 61690 y Fo(Z)63 b Fh(2)p Fo(\032)p Fp(\(Supp)155 b Fo( )32 b Fp(\))27940 60074 y Fr(S)77 b Fs(\()p Fq(S)30127 60273 y Fo(Z)30883 60074 y Fs(\))31389 57351 y Fj(9)31389 58547 y(=)31389 60938 y(;)32938 60074 y Fs(=)369 b(1)p Fr(=)-72 b(A)36522 60273 y Fo( )37594 60074 y Fr(>)368 b Fs(0)p Fr(:)9703 b Fs(\(71\))0 63750 y(No)-36 b(w)434 b(let)13411 65612 y Fr(Q)14441 65811 y Fp(0)15336 65612 y Fs(:=)369 b Fq(f)p Fr(W)550 b Fs(:)369 b Fr(Q)p Fs([)p Fr(W)181 b Fs(])369 b Fq(\024)g Fs(1)p Fr(;)221 b Fq(j)p Fr(W)181 b Fq(j)370 b(\024)f Fr(A)31308 65811 y Fo( )32010 65612 y Fq(g)g(\033)g Fs(Supp)220 b Fr(g)48 b(:)11099 b Fs(\(72\))1328 67745 y(Appro)-36 b(ximating)434 b(sums)f(b)-36 b(y)434 b(in)-36 b(tegrals,)434 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)6991 70777 y Fr(L)7876 70228 y Fh(\000)p Fo(b)9011 70351 y Fg(3)9527 70777 y Fq(N)10617 70976 y Fo( )11319 70777 y Fs(\()p Fr(L)p Fs(\))368 b(=)h Fr(L)15850 70228 y Fh(\000)p Fo(b)16985 70351 y Fg(3)18435 69515 y Fj(X)17722 72485 y Fo(N)94 b Fh(2)p Fi(Z)19780 72190 y Fg(2)p Ff(b)20550 72389 y Fg(3)21288 70777 y Fr(g)48 b Fs(\()p Fr(N)30 b(=)24187 69587 y Fq(p)p 25294 69587 V 1190 x Fr(L)p Fs(\))369 b(=)28434 68969 y Fj(Z)29172 71976 y Fi(R)29812 71681 y Fg(2)p Ff(b)30582 71880 y Fg(3)31375 70777 y Fr(g)48 b Fs(\()p Fr(W)181 b Fs(\))221 b Fr(dW)474 b Fs(+)39086 69515 y Fj(X)38373 72485 y Fo(N)94 b Fh(2)p Fi(Z)40431 72190 y Fg(2)p Ff(b)41201 72389 y Fg(3)41940 70777 y Fr(E)42903 70976 y Fo(N)42 b(;L)44648 70777 y Fr(;)p eop %%Page: 25 25 25 24 bop 11424 681 a Fm(CRITICAL)440 b(POINTS)j(AND)g(SUPERSYMMETRIC)e (V)-114 b(A)-28 b(CUA,)443 b(I)28 b(I)g(I)10843 b(25)0 3337 y Fs(where)5032 6000 y Fr(E)5995 6199 y Fo(N)42 b(;L)8847 6000 y Fs(=)10966 4192 y Fj(Z)11704 7200 y Fh(R)12507 7349 y Ff(N)29 b(;L)14103 6000 y Fs([)p Fr(g)48 b Fs(\()p Fr(N)30 b(=)17363 4811 y Fq(p)p 18470 4811 885 54 v 1189 x Fr(L)p Fs(\))295 b Fq(\000)g Fr(g)48 b Fs(\()p Fr(W)181 b Fs(\)])221 b Fr(dW)-36 b(;)10966 9272 y Fq(R)12092 9471 y Fo(N)42 b(;L)14206 9272 y Fs(=)369 b Fq(f)p Fr(W)549 b Fs(=)369 b(\()p Fr(W)21143 9471 y Fp(1)21669 9272 y Fr(;)221 b(:)g(:)g(:)i(;)e(W)25809 9471 y Fp(2)p Fo(b)26682 9594 y Fg(3)27200 9272 y Fs(\))368 b Fq(2)h Fk(R)30288 8723 y Fp(2)p Fo(b)31161 8846 y Fg(3)32047 9272 y Fs(:)g Fr(N)33819 9471 y Fo(j)34674 9272 y Fr(<)g(W)37283 9471 y Fo(j)38139 9272 y Fr(<)g(N)40562 9471 y Fo(j)41343 9272 y Fs(+)295 b(1)p Fr(=)43950 8082 y Fq(p)p 45058 8082 V 45058 9272 a Fr(L)p Fq(g)p Fr(:)1328 11565 y Fs(Let)17575 13317 y Fr(B)436 b Fs(=)369 b Fr(Q)21410 13516 y Fp(0)22231 13317 y Fq(\\)295 b Fs([)p Fr(@)72 b(Q)295 b Fq([)g(D)332 b([)294 b Fr(\031)48 b Fs(\()p Fr(@)72 b(K)95 b Fs(\)])369 b Fr(:)0 15340 y Fs(Since)388 b Fr(g)435 b Fs(is)389 b(lo)36 b(cally)390 b(constan)-36 b(t)388 b(on)g Fq(S)302 b Fk(r)202 b Fr(B)67 b Fs(,)398 b(the)388 b(error)g Fr(E)27154 15539 y Fo(N)42 b(;L)29287 15340 y Fs(v)-72 b(anishes)388 b(whenev)-36 b(er)388 b Fq(R)41139 15539 y Fo(N)42 b(;L)43087 15340 y Fq(\\)202 b Fr(B)436 b Fs(=)369 b Fq(;)p Fs(.)563 b(Hence)3045 16864 y Fj(X)2332 19834 y Fo(N)94 b Fh(2)p Fi(Z)4390 19539 y Fg(2)p Ff(b)5160 19738 y Fg(3)5899 18126 y Fr(E)6862 18325 y Fo(N)42 b(;L)8976 18126 y Fq(\024)369 b Fs(\(sup)220 b Fr(f)142 b Fs(\))p Fr(L)15237 17578 y Fh(\000)p Fo(b)16372 17701 y Fg(3)16888 17050 y Fj(\002)17442 18126 y Fs(#)p Fq(f)p Fr(N)507 b Fs(:)369 b Fq(R)22595 18325 y Fo(N)42 b(;L)24636 18126 y Fq(\\)294 b Fr(B)436 b Fq(6)p Fs(=)369 b Fq(;g)29949 17050 y Fj(\003)30872 18126 y Fs(=)f Fr(L)33137 17578 y Fh(\000)p Fo(b)34272 17701 y Fg(3)35010 18126 y Fr(O)36261 16652 y Fj(\020)37055 16937 y Fq(p)p 38162 16937 V 1189 x Fr(L)39046 17181 y Fp(2)p Fo(b)39919 17304 y Fg(3)40381 17181 y Fh(\000)p Fp(1)41638 16652 y Fj(\021)42801 18126 y Fs(=)g Fr(O)36 b Fs(\()p Fr(L)46602 17578 y Fh(\000)p Fp(1)p Fo(=)p Fp(2)48801 18126 y Fs(\))p Fr(:)50967 21774 y Fc(\003)0 24502 y Fs(4.1.1.)652 b Fl(Non-clustering)522 b(of)i(critic)-66 b(al)522 b(p)-66 b(oints.)649 b Fs(Before)499 b(concluding)e(the)g(pro) 36 b(of)499 b(of)f(Prop)36 b(osition)499 b(1.3,)515 b(w)-36 b(e)0 26052 y(brie\015y)290 b(consider)f(the)g(question)h(of)h(whether) e(there)g(exist)h(real)g(h)-36 b(yp)36 b(ersurfaces)290 b(\000)369 b Fq(\032)g(C)f Fs(with)289 b(the)g(prop)36 b(ert)-36 b(y)0 27917 y(that)348 b Fq(\030)4135 26792 y(p)p 5242 26792 V 1125 x Fr(L)6127 27036 y Fp(2)p Fo(b)7000 27159 y Fg(3)7461 27036 y Fh(\000)p Fp(1)9067 27917 y Fs(critical)i(p)36 b(oin)-36 b(ts)348 b(of)h(norm)g Fq(\024)369 b Fr(L)348 b Fs(cluster)g(within)h(a)g(1)p Fr(=L)g Fs(tub)36 b(e)348 b(around)f(\000.)550 b(A)349 b(domain)f(in)0 29466 y Fq(C)429 b Fs(whose)352 b(b)36 b(oundary)351 b(con)-36 b(tained)351 b(a)h(piece)f(of)h(\000)g(w)-36 b(ould)351 b(attain)h(the)f(remainder)f(estimate)i(in)f(Prop)36 b(osition)0 31016 y(1.3.)1328 32566 y(Since)461 b(the)g(n)-36 b(um)g(b)36 b(er)459 b(of)j(critical)g(p)36 b(oin)-36 b(ts)461 b(corresp)36 b(onding)461 b(to)g Fr(G)416 b Fq(2)f Fr(H)35511 32084 y Fp(3)36036 32566 y Fs(\()p Fr(X)32 b(;)221 b Fk(Z)315 b Fq(\010)40783 31522 y(p)p 41890 31522 1684 54 v 1044 x(\000)p Fs(1)p Fk(Z)p Fs(\))461 b(is)g(b)36 b(ounded,)0 34115 y(suc)-36 b(h)523 b(clustering)h(of)h (critical)g(p)36 b(oin)-36 b(ts)524 b(could)g(only)g(o)36 b(ccur)524 b(if)h(a)f(sublattice)g(of)h(rank)f(2)p Fr(b)43487 34314 y Fp(3)44370 34115 y Fq(\000)357 b Fs(1)525 b(clustered)0 35665 y(around)433 b(the)g(h)-36 b(yp)36 b(ersurface)20205 36369 y Fj([)19341 39247 y Fp(\()p Fo(z)h(;\034)102 b Fp(\))p Fh(2)p Fp(\000)22766 37631 y Fr(H)23947 37082 y Fp(3)23844 37959 y Fo(z)37 b(;\034)25524 37631 y Fq(\032)369 b Fr(H)28107 37082 y Fp(3)28632 37631 y Fs(\()p Fr(X)32 b(;)221 b Fk(C)p Fs(\))p Fr(:)17031 b Fs(\(73\))1328 41238 y(There)327 b(do)g(exist)h(real)g(h)-36 b(yp)36 b(ersurfaces)327 b(in)g Fr(H)22667 40756 y Fp(3)23192 41238 y Fs(\()p Fr(X)32 b(;)221 b Fk(C)p Fs(\))328 b(for)g(whic)-36 b(h)326 b(suc)-36 b(h)327 b(exceptional)h(clustering)f(o)36 b(ccurs,)0 42787 y(namely)460 b(h)-36 b(yp)36 b(erplanes)460 b(con)-36 b(taining)460 b(a)g(sublattice)f(of)i(rank)f(2)p Fr(b)30919 42986 y Fp(3)31758 42787 y Fq(\000)313 b Fs(1.)658 b(W)-108 b(e)460 b(refer)g(to)g(suc)-36 b(h)458 b(a)j(h)-36 b(yp)36 b(erplane)0 44337 y(as)431 b(a)g(rational)h(h)-36 b(yp)36 b(erplane)430 b Fr(L)p Fs(.)577 b(F)-108 b(or)431 b(instance,)g(an)-36 b(y)431 b(pair)g(of)g(in)-36 b(tegral)431 b(cycles)h Fr(\015)39504 44536 y Fp(1)40030 44337 y Fr(;)221 b(\015)41287 44536 y Fp(2)42243 44337 y Fs(de\014nes)430 b(a)h(rational)0 45887 y(h)-36 b(yp)36 b(erplane)4913 48808 y Fr(L)369 b Fs(=)f Fr(L)8432 49007 y Fo(\015)8917 49130 y Fg(1)9378 49007 y Fo(;\015)10124 49130 y Fg(2)11010 48808 y Fs(=)g Fq(f)p Fr(G)i Fs(=)e Fr(F)476 b Fs(+)295 b Fr(iH)471 b Fq(2)368 b Fr(H)22883 48260 y Fp(3)23408 48808 y Fs(\()p Fr(X)32 b(;)221 b Fk(C)p Fs(\))371 b(:)e Fr(`)p Fs(\()p Fr(F)475 b Fs(+)295 b Fr(iG)p Fs(\))369 b(:=)35936 47000 y Fj(Z)36674 50007 y Fo(\015)37159 50130 y Fg(1)37896 48808 y Fr(F)476 b Fs(+)40521 47000 y Fj(Z)41259 50007 y Fo(\015)41744 50130 y Fg(2)42481 48808 y Fr(H)c Fs(=)368 b(0)p Fq(g)p Fr(:)0 52184 y Fs(As)569 b(men)-36 b(tioned)569 b(in)g(the)g(in)-36 b(tro)36 b(duction,)602 b(pro)72 b(jections)570 b(of)g(the)f(lattice)h(p)36 b(oin)-36 b(ts)569 b Fr(H)40674 51702 y Fp(3)41199 52184 y Fs(\()p Fr(X)32 b(;)221 b Fk(Z)388 b Fq(\010)46093 51141 y(p)p 47200 51141 V 1043 x(\000)p Fs(1)p Fk(Z)p Fs(\))569 b(to)0 54122 y Fr(@)72 b(Q)539 b Fs(concen)-36 b(trate)538 b(to)h(sub-leading) f(order)21580 52998 y Fq(p)p 22687 52998 885 54 v 1124 x Fr(L)23572 53242 y Fp(2)p Fo(b)24445 53365 y Fg(3)24906 53242 y Fh(\000)p Fp(1)26702 54122 y Fs(around)g(the)h(h)-36 b(yp)36 b(ersurface)538 b(of)i Fr(@)72 b(Q)539 b Fs(obtained)f(b)-36 b(y)0 55672 y(in)g(tersecting)433 b(it)h(with)g(a)f(rational)i(h)-36 b(yp)36 b(erplane.)1328 57222 y(Ho)-36 b(w)g(ev)g(er,)342 b(rational)319 b(h)-36 b(yp)36 b(erplanes)317 b(nev)-36 b(er)317 b(ha)-36 b(v)g(e)318 b(the)f(form)h(\(73\).)540 b(Indeed,)340 b(under)317 b(the)g(corresp)36 b(ondence)0 58771 y Fr(\032)347 b Fq(\016)h Fr(\031)2816 58289 y Fh(\003)3851 58771 y Fs(de\014ned)509 b(b)-36 b(y)510 b(the)g(diagram)h(\(60\),)530 b(the)510 b(image)h(of)g(a)g(h)-36 b(yp)36 b(erplane)509 b(alw)-36 b(a)g(ys)512 b(co)-36 b(v)g(ers)511 b(a)g(region)f(and)0 60321 y(not)433 b(a)h(h)-36 b(yp)36 b(ersurface)434 b(of)g Fq(C)77 b Fs(.)579 b(That)434 b(is,)17231 62615 y(dim\()p Fr(L)295 b Fq(\\)f Fr(H)23446 62067 y Fp(3)23343 62944 y Fo(z)37 b(;\034)24655 62615 y Fs(\))369 b Fr(>)f Fs(1)738 b Fq(8)p Fs(\()p Fr(z)59 b(;)221 b(\034)148 b Fs(\))370 b Fq(2)f(C)77 b Fr(:)0 65023 y Fs(Indeed,)468 b(under)460 b(the)h(iden)-36 b(ti\014cation)462 b Fr(H)19654 64541 y Fp(3)19551 65352 y Fo(z)37 b(;\034)21279 65023 y Fq(')416 b Fr(H)23909 64541 y Fp(3)24435 65023 y Fs(\()p Fr(X)32 b(;)221 b Fk(R)p Fs(\))p Fr(;)463 b(L)p Fq(j)30177 65246 y Fo(H)31018 64994 y Fg(3)30950 65467 y Ff(z)26 b(;\034)32660 65023 y Fs(b)36 b(ecomes)461 b(the)h(real)g(linear)g(functional)0 66777 y Fr(L)p Fs(\()p Fr(H)103 b Fs(\))552 b(=)5194 65707 y Fj(R)5822 67238 y Fo(\015)6307 67361 y Fg(1)7044 66777 y Fr(F)181 b Fs(\()p Fr(z)59 b(;)221 b(\034)76 b(;)221 b(H)103 b Fs(\))369 b(+)14478 65707 y Fj(R)15105 67238 y Fo(\015)15590 67361 y Fg(2)16327 66777 y Fr(H)644 b Fs(on)542 b Fr(H)21145 66295 y Fp(3)21670 66777 y Fs(\()p Fr(X)32 b(;)221 b Fk(R)p Fs(\).)904 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3337 y Fs(nev)-36 b(er)504 b(ha)-36 b(v)g(e)505 b(this)f(form)g(is)h(strong)f(evidence)g (that)g(there)g(are)g(no)g(smo)36 b(oth)505 b(h)-36 b(yp)36 b(ersurfaces)504 b(\000)489 b Fq(\032)g(C)582 b Fs(for)0 4887 y(whic)-36 b(h)433 b(lattice)i(p)36 b(oin)-36 b(ts)433 b(cluster)g(to)h(subleading)f(order)g(around)g(\(73\).)0 7383 y(4.2.)1294 b FA(Hessians)435 b(and)e(densit)-42 b(y)433 b(of)g(critical)h(p)42 b(oin)-42 b(ts.)651 b Fs(The)377 b(\014nal)f(step)g(in)g(the)g(pro)36 b(of)376 b(of)h(Prop)36 b(osition)0 8933 y(1.3)435 b(is)e(to)h(c)-36 b(hange)434 b(the)f(order)g(of)h(in)-36 b(tegration)434 b(o)-36 b(v)g(er)434 b Fq(C)511 b Fs(and)433 b(o)-36 b(v)g(er)434 b Fq(S)33129 9132 y Fo(Z)33885 8933 y Fs(:)0 11307 y Ft(Lemma)497 b FA(4.2.)554 b Fl(We)465 b(have:)3597 12045 y Fj(Z)4335 15053 y Fh(f)p Fo(Q)p Fp([)p Fo(W)131 b Fp(])p Fh(\024)p Fp(1)p Fh(g)8814 13854 y Fq(h)p Fr(C)10262 14053 y Fo(W)11335 13854 y Fr(;)221 b( )48 b Fq(i)221 b Fr(dW)550 b Fs(=)17383 12045 y Fj(Z)18121 15053 y Fh(C)18944 12045 y Fj(Z)19681 15053 y Fh(S)20251 15202 y Ff(Z)21186 13854 y Fr( )48 b Fs(\()p Fr(Z)23 b(;)221 b(W)181 b Fs(\))221 b Fq(j)g Fs(det)g Fr(H)30016 13305 y Fo(c)30478 13854 y Fr(W)181 b Fs(\()p Fr(Z)95 b Fs(\))p Fq(j)221 b Fr(\037)35289 14053 y Fo(Q)36029 14202 y Ff(Z)36742 13854 y Fs(\()p Fr(W)181 b Fs(\))221 b Fr(dW)402 b(d)p Fs(V)-108 b(ol)44244 14053 y Fo(W)131 b(P)46045 13854 y Fs(\()p Fr(Z)95 b Fs(\))p Fr(:)1328 17131 y Fs(Com)-36 b(bining)567 b(the)g(form)-36 b(ulas)567 b(in)g(Lemmas)g(4.1)h(and)e(4.2,)602 b(w)-36 b(e)567 b(obtain)g(the)f(form)-36 b(ula)568 b(of)f(Prop)36 b(osition)0 18681 y(\(1.3\))q(.)1328 20231 y(The)478 b(pro)36 b(of)478 b(of)h(Lemma)f(4.2)g(is)g(in)g(t)-36 b(w)g(o)478 b(parts.)710 b(The)478 b(\014rst)f(is)h(an)g(elemen)-36 b(tary)478 b(exercise)g(in)g(c)-36 b(hanging)0 21780 y(v)-72 b(ariables)459 b(in)f(an)h(in)-36 b(tegral,)465 b(whic)-36 b(h)458 b(w)-36 b(e)458 b(accomplish)h(b)36 b(elo)-36 b(w)459 b(b)-36 b(y)458 b(relating)h(b)36 b(oth)458 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Fh(f)p Fo(Q)p Fp([)p Fo(W)131 b Fp(])p Fh(\024)p Fp(1)p Fh(g)30360 35122 y Fq(h)p Fr(C)31808 35321 y Fo(W)32881 35122 y Fr(;)221 b( )48 b Fq(i)221 b Fr(dW)-36 b(:)12364 b Fs(\(74\))1328 38204 y(The)623 b(measure)f Fr(d\026)10904 38403 y Fh(I)12160 38204 y Fs(ma)-36 b(y)624 b(b)36 b(e)622 b(expressed)g(in)h(terms)f(of)h(the)f(Lera)-36 b(y)623 b(measure)f Fr(d)p Fq(L)43428 38403 y Fh(I)44685 38204 y Fs(de\014ned)f(b)-36 b(y)623 b(a)0 39754 y(measure)433 b Fr(d\027)517 b Fs(on)434 b Fq(S)533 b Fs(and)433 b(the)g(`ev)-72 b(aluation)435 b(map')17254 41931 y Fr(")368 b Fs(:)i(\()p Fr(Z)23 b(;)221 b(W)181 b Fs(\))369 b Fq(2)f(C)374 b(\002)295 b(S)469 b(!)369 b(r)p Fr(W)181 b Fs(\()p Fr(Z)95 b Fs(\))p Fr(:)0 44178 y Fs(The)434 b(Lera)-36 b(y)433 b(form)h(is)g(the)f (quotien)-36 b(t)434 b Fr(d)p Fq(L)19646 44377 y Fh(I)20649 44178 y Fs(:=)22524 43617 y Fo(dV)23556 43766 y Ff(W)108 b(P)25109 43617 y Fh(\002)p Fo(d\027)p 22524 43872 4322 54 v 24225 44636 a(d")26978 44178 y Fs(,)434 b(i.e.)579 b(the)433 b(unique)g(form)h(satisfying)19283 46356 y Fr(d)p Fq(L)20875 46555 y Fh(I)21804 46356 y Fq(\002)296 b Fr(d")368 b Fs(=)h Fr(dV)27601 46555 y Fo(W)131 b(P)29697 46356 y Fq(\002)296 b Fr(d\027)12 b(:)0 48534 y Fs(This)434 b(measure)f(is)h(often)g(written)f Fr(\016)50 b Fs(\()p Fq(r)p Fr(W)181 b Fs(\()p Fr(Z)95 b Fs(\)\))p Fr(dW)181 b(dV)288 b Fs(\()p Fr(Z)95 b Fs(\))434 b(in)f(the)g(ph)-36 b(ysics)434 b(literature.)1328 50083 y(As)375 b(suggested)f(b)-36 b(y)375 b(the)e(ph)-36 b(ysics)375 b(form)-36 b(ula,)387 b Fr(d\026)23905 50282 y Fh(I)24907 50083 y Fs(=)369 b Fq(r)p Fr(s)p Fs(\()p Fr(Z)95 b Fs(\))30005 49601 y Fh(\003)30531 50083 y Fr(\016)31107 50282 y Fp(0)31633 50083 y Fr(:)374 b Fs(Ho)-36 b(w)g(ev)g(er,)388 b(this)374 b(form)-36 b(ula)375 b(is)g(somewhat)0 51633 y(am)-36 b(biguous.)662 b(If)462 b(w)-36 b(e)461 b(regard)h Fr(s)f Fs(as)g(\014xed,)469 b(then)460 b(it)i(is)f(simply)h(the)f(pullbac)-36 b(k)462 b(of)g Fr(\016)40075 51832 y Fp(0)41062 51633 y Fs(under)e 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y(dimensional)391 b(manifold)h Fr(M)530 b Fs(and)390 b(a)h(space)g Fq(S)491 b Fs(of)392 b(functions)f Fr(F)549 b Fs(:)370 b Fr(M)32886 70880 y Fo(n)33881 70681 y Fq(!)f Fk(R)36537 70199 y Fo(n)37164 70681 y Fs(.)564 b(In)391 b(our)g(case,)400 b Fr(F)549 b Fs(=)369 b Fq(r)p Fr(s)390 b Fs(and)0 72231 y Fr(M)563 b Fs(is)423 b(a)h(co)36 b(ordinate)424 b(neigh)-36 b(b)36 b(orho)g(o)g(d)423 b(in)h Fq(C)501 b Fs(where)424 b Fr(M)562 b Fs(has)424 b(lo)36 b(cal)425 b(co)36 b(ordinates)423 b(\()p Fr(x)40521 72430 y Fp(1)41047 72231 y Fr(;)221 b(:)g(:)g(:)j(;)d(x)44699 72430 y Fo(n)45326 72231 y Fs(\))423 b(and)g Fq(L)h Fs(has)p eop %%Page: 27 27 27 26 bop 11424 681 a Fm(CRITICAL)440 b(POINTS)j(AND)g(SUPERSYMMETRIC)e (V)-114 b(A)-28 b(CUA,)443 b(I)28 b(I)g(I)10843 b(27)0 3337 y Fs(a)415 b(lo)36 b(cal)417 b(frame.)572 b(Supp)36 b(ose)414 b(that)h(0)g(is)g(a)h(regular)f(v)-72 b(alue)415 b(of)h Fr(F)181 b Fs(,)418 b(so)e(that)e Fr(F)596 b Fs(is)415 b(a)g(lo)36 b(cal)417 b(di\013eomorphism)d(in)0 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Fr(d)507 b Fs(=)f(2)p Fr(b)25654 11449 y Fp(3)26687 11250 y Fr(>)h(n)g Fs(=)g Fr(b)31561 11449 y Fp(3)32086 11250 y Fs(,)535 b(so)515 b(w)-36 b(e)515 b(in)-36 b(tro)36 b(duce)514 b(a)h(supplemen)-36 b(tary)0 12800 y(linear)462 b(map:)634 b(for)463 b(a)f(p)36 b(oin)-36 b(t)461 b Fr(u)417 b Fq(2)f Fr(U)556 b Fq(\032)417 b Fr(M)139 b Fs(,)469 b Fq(S)22024 12999 y Fo(u)23086 12800 y Fs(is)462 b(the)f(k)-36 b(ernel)462 b(of)g Fr(")32666 12999 y Fo(u)33266 12800 y Fs(,)469 b(and)461 b(w)-36 b(e)462 b(supplemen)-36 b(t)460 b Fr(")46141 12999 y Fo(u)47203 12800 y Fs(with)i(the)0 14350 y(pro)72 b(jection)434 b(\005)7153 14549 y Fo(u)8122 14350 y Fs(:)370 b Fq(S)469 b(!)369 b(S)12627 14549 y Fo(u)13227 14350 y Fr(:)434 b Fs(Then,)19469 16694 y(\()p Fr(")20584 16893 y Fo(u)21184 16694 y Fr(;)221 b Fs(\005)22741 16893 y Fo(u)23342 16694 y Fs(\))368 b(:)i Fq(S)469 b(!)369 b Fk(R)28876 16146 y Fo(n)29798 16694 y Fq(\010)295 b(S)31930 16893 y Fo(u)0 19039 y Fs(is)434 b(a)g(linear)g(isomorphism.)578 b(Hence)434 b(it)f(equals)i(its)e(deriv)-72 b(ativ)-36 b(e,)435 b(so)12508 21361 y Fq(h)p Fr(")13634 20879 y Fh(\003)13634 21689 y Fo(u)14233 21361 y Fr(\016)14809 21560 y Fp(0)15335 21361 y Fr(;)221 b(')p Fq(i)1107 b Fs(=)g Fq(h)p Fr(\016)21605 21560 y Fp(0)22130 21361 y Fr(;)221 b(')p Fs(\(\()p Fr(")25185 21560 y Fo(u)25785 21361 y Fr(;)g Fs(\005)27342 21560 y Fo(u)27943 21361 y Fs(\))28449 20879 y Fh(\000)p Fp(1)29707 21361 y Fs(\))p Fq(j)g Fs(det)o(\()p Fr(")33724 21560 y Fo(u)34324 21361 y Fr(;)g Fs(\005)35881 21560 y Fo(u)36482 21361 y Fs(\))36988 20879 y Fh(\000)p Fp(1)38245 21361 y Fq(ji)p Fr(:)0 23728 y Fs(No)-36 b(w,)417 b Fq(S)512 b Fs(is)412 b(equipp)36 b(ed)411 b(with)h(an)g(inner)f(pro)36 b(duct,)416 b(whic)-36 b(h)411 b(induces)g(an)h(inner)f(pro)36 b(duct)411 b(on)h Fk(R)45696 23246 y Fo(n)46573 23728 y Fq(\010)251 b(S)48661 23927 y Fo(u)49262 23728 y Fs(.)571 b(W)-108 b(e)0 25277 y(c)-36 b(ho)36 b(ose)378 b(an)f(orthonormal)g(basis)g Fq(f)p Fr(S)17807 25476 y Fp(1)18333 25277 y Fr(;)221 b(:)g(:)g(:)j(;)d(S)22046 25476 y Fo(n)22673 25277 y Fq(g)377 b Fs(of)h Fq(S)26043 24795 y Fh(?)25943 25606 y Fo(u)26830 25277 y Fr(;)g Fs(and)e Fq(f)p Fr(S)31505 25476 y Fo(n)p Fp(+1)33334 25277 y Fr(;)221 b(:)g(:)g(:)j(;)d(S)37047 25476 y Fo(d)37587 25277 y Fq(g)377 b Fs(for)h Fq(S)41363 25476 y Fo(u)41963 25277 y Fs(.)560 b(Since)377 b(\005)47199 25476 y Fo(u)48168 25277 y Fs(:)369 b Fq(S)49702 25476 y Fo(u)50672 25277 y Fq(!)0 26827 y(S)804 27026 y Fo(u)1825 26827 y Fs(is)421 b(the)g(iden)-36 b(tit)g(y)-108 b(,)423 b(\()p Fr(")11511 27026 y Fo(u)12110 26827 y Fr(;)221 b Fs(\005)13667 27026 y Fo(u)14268 26827 y Fs(\))421 b(has)g(a)g(blo)36 b(c)-36 b(k)421 b(diagonal)h(matrix)g(relativ)-36 b(e)421 b(to)g(the)g(bases)g(of)g Fq(S)469 b Fs(=)369 b Fq(S)48237 26345 y Fh(?)48137 27156 y Fo(u)49293 26827 y Fq(\010)269 b(S)51399 27026 y Fo(u)0 28377 y Fs(and)487 b Fk(R)3542 27895 y Fo(n)4500 28377 y Fq(\010)332 b(S)6669 28576 y Fo(u)7270 28377 y Fs(,)501 b(with)488 b(the)e(iden)-36 b(tit)g(y)487 b(in)h(the)f Fq(S)22971 28576 y Fo(u)23571 28377 y Fs(-)p Fq(S)24809 28576 y Fo(u)25897 28377 y Fs(blo)36 b(c)-36 b(k.)740 b(Hence,)501 b(det\()p Fr(")37212 28576 y Fo(u)37811 28377 y Fr(;)221 b Fs(\005)39368 28576 y Fo(u)39969 28377 y Fs(\))461 b(=)f(det)o(\()p Fr(")45329 28576 y Fo(u)45929 28377 y Fq(j)46298 28633 y Fh(S)46933 28381 y Fe(?)47683 28377 y Fs(\))487 b(where)0 29927 y(the)433 b(determinan)-36 b(t)432 b(is)i(with)g(resp)36 b(ect)433 b(to)h(these)f(bases.)1328 31476 y(The)h(general)g(case)g(of) g(form)-36 b(ula)434 b(\(74\))g(states)g(that)15955 34535 y Fr(d\026)17414 34734 y Fh(I)18416 34535 y Fs(=)369 b Fq(j)221 b Fs(det)g Fr(D)36 b(W)181 b Fs(\()p Fr(u)p Fs(\))p Fq(j)295 b(\002)28817 33636 y Fr(\037)29632 33835 y Fo(Q)30427 33636 y Fr(du)g Fq(\002)h Fr(dW)p 28817 34230 6735 54 v 31542 35446 a(d")35684 34535 y(:)13643 b Fs(\(76\))0 37274 y(W)-108 b(e)429 b(then)g(compute)f(the)h Fq(I)527 b Fs(in)-36 b(tegral)430 b(as)g(an)f(iterated)g(in)-36 b(tegral)429 b(using)h(the)e(other)h(singular)h(\014bration)e Fr(\031)48 b Fs(,)0 38824 y(i.e.)579 b(b)-36 b(y)434 b(\014rst)e(in)-36 b(tegrating)434 b(o)-36 b(v)g(er)434 b(the)f(\014b)36 b(ers)433 b Fq(S)22677 39023 y Fo(u)23277 38824 y Fs(:)8263 40075 y Fj(Z)9001 43082 y Fh(I)9856 41883 y Fr( )48 b Fs(\()p Fr(u)p Fs(\))p Fr(d\026)13961 42082 y Fh(I)14963 41883 y Fs(=)16343 40075 y Fj(Z)17081 43082 y Fo(U)18089 40075 y Fj(Z)18827 43082 y Fh(S)19397 43193 y Ff(u)22399 40984 y Fr( )g Fs(\()p Fr(u)p Fs(\))p 20351 41577 6741 54 v 20351 42794 a Fq(j)221 b Fs(det\()p Fr(")23863 42993 y Fo(u)24463 42794 y Fq(j)24832 43050 y Fh(S)25467 42798 y Fe(?)25402 43271 y Ff(u)26217 42794 y Fs(\))p Fq(j)27224 41883 y Fr(\037)28039 42082 y Fo(Q)28779 42193 y Ff(u)29380 41883 y Fs(\()p Fr(W)181 b Fs(\))p Fq(j)221 b Fs(det)f Fr(D)36 b(W)181 b Fs(\()p Fr(u)p Fs(\))p Fq(j)221 b Fr(dW)402 b(du)369 b(:)5951 b Fs(\(77\))1328 45173 y(Returning)433 b(to)h(our)f(case)h(where)g Fr(F)549 b Fs(=)369 b Fq(r)p Fr(s)p Fs(,)433 b(\(77\))h(b)36 b(ecomes)5262 46497 y Fj(Z)6000 49505 y Fh(I)6855 48305 y Fr( )48 b Fs(\()p Fr(Z)95 b Fs(\))p Fr(d\026)11205 48504 y Fh(I)12207 48305 y Fs(=)13588 46497 y Fj(Z)14326 49505 y Fh(C)15148 46497 y Fj(Z)15886 49505 y Fh(S)16456 49654 y Ff(Z)18566 47407 y Fr( )48 b Fs(\()p Fr(Z)23 b(;)221 b(W)181 b Fs(\))p 17523 48000 6896 54 v 17523 49217 a Fq(j)221 b Fs(det\()p Fr(")21035 49416 y Fo(Z)21790 49217 y Fq(j)22159 49525 y Fh(S)22794 49220 y Fe(?)22729 49833 y Ff(Z)23544 49217 y Fs(\))p Fq(j)24552 48305 y(j)g Fs(det)g Fr(H)28351 47757 y Fo(c)28813 48305 y Fr(W)181 b Fs(\()p Fr(Z)95 b Fs(\))p Fq(j)221 b Fr(\037)33624 48504 y Fo(Q)34364 48653 y Ff(Z)35077 48305 y Fs(\()p Fr(W)181 b Fs(\))221 b Fr(dW)402 b(d)p Fs(V)-108 b(ol)42579 48504 y Fo(W)131 b(P)44380 48305 y Fs(\()p Fr(Z)95 b Fs(\))p Fr(:)2950 b Fs(\(78\))0 52146 y(4.3.)652 b FA(Completion)521 b(of)e(the)g(pro)42 b(of)520 b(of)f(Lemma)h(4.2.)651 b Fs(T)-108 b(o)453 b(complete)f(the)f(pro)36 b(of)453 b(of)g(the)e(lemma,)458 b(w)-36 b(e)0 53696 y(need)337 b(to)g(sho)-36 b(w)338 b(that)f Fq(j)221 b Fs(det)o(\()p Fr(")13792 53895 y Fo(Z)14547 53696 y Fq(j)14916 54005 y Fh(S)15551 53700 y Fe(?)15486 54313 y Ff(Z)16301 53696 y Fs(\))p Fq(j)369 b Fs(=)g(1)337 b(with)h(resp)36 b(ect)337 b(to)g(normal)h(co)36 b(ordinates)338 b(and)f(an)g(adapted)f(frame)0 55420 y(at)434 b Fr(Z)2480 55619 y Fp(0)3375 55420 y Fs(=)368 b(\()p Fr(z)5865 55619 y Fp(0)6391 55420 y Fr(;)221 b(\034)7538 55619 y Fp(0)8065 55420 y Fs(\))368 b Fq(2)h Fr(M)139 b Fs(.)1328 56970 y(Recalling)453 b(\(65\))q({\(66\),)434 b(w)-36 b(e)434 b(write)16165 59167 y Fj(e)15897 59502 y Fq(S)16801 58954 y Fh(?)16701 59831 y Fo(Z)17958 59502 y Fs(=)368 b Fr(I)19912 59701 y Fo(\034)20782 59502 y Fq(\016)296 b(W)23163 58954 y Fh(\000)p Fp(1)24421 59502 y Fs(\()p Fq(S)25831 58954 y Fh(?)25731 59831 y Fo(Z)26618 59502 y Fs(\))368 b(=)h Fr(H)30054 58954 y Fp(3)p Fo(;)p Fp(0)29951 59831 y Fo(z)31606 59502 y Fq(\010)295 b Fr(H)34115 58954 y Fp(1)p Fo(;)p Fp(2)34012 59831 y Fo(z)35741 59502 y Fr(:)0 62035 y Fs(A)434 b(complex)h(orthonormal)f(basis)g(for)19396 61699 y Fj(e)19129 62035 y Fq(S)20033 61553 y Fh(?)19933 62371 y Fo(Z)20571 62494 y Fg(0)21521 62035 y Fs(relativ)-36 b(e)435 b(to)f Fr(Q)g Fs(is)h Fq(f)155 b Fs(\026)-805 b Fr(\037)31988 62234 y Fp(0)32513 62035 y Fr(;)377 b Fs(\026)-806 b Fr(\037)33910 62234 y Fp(1)34437 62035 y Fr(;)221 b(:)g(:)g(:)i(;)377 b Fs(\026)-806 b Fr(\037)38164 62258 y Fo(h)38706 62006 y Fg(2)p Ff(;)p Fg(1)39882 62035 y Fq(g)p Fs(,)434 b(where)g Fr(\037)45914 62234 y Fp(0)46809 62035 y Fs(=)p 48191 60968 940 54 v 370 w(\012)49130 62234 y Fo(z)49568 62357 y Fg(0)50084 62035 y Fs(.)580 b(A)0 63701 y(basis)434 b(\(o)-36 b(v)g(er)434 b Fk(R)p Fs(\))f(for)i Fq(S)11272 63219 y Fh(?)11172 64037 y Fo(Z)11810 64160 y Fg(0)12760 63701 y Fs(is)p 7415 65213 1023 54 v 7415 66280 a Fr(U)8437 66479 y Fo(j)9293 66280 y Fs(:=)369 b Fq(W)405 b(\016)295 b Fr(I)14389 65732 y Fh(\000)p Fp(1)14285 66609 y Fo(\034)15647 66280 y Fs(\()155 b(\026)-805 b Fr(\037)16968 66479 y Fo(j)17454 66280 y Fs(\))p Fr(;)p 20277 65213 1048 54 v 1956 w(V)21324 66479 y Fo(j)22180 66280 y Fs(:=)369 b Fq(W)405 b(\016)295 b Fr(I)27276 65732 y Fh(\000)p Fp(1)27172 66609 y Fo(\034)28534 66280 y Fs(\()29040 65172 y Fq(p)p 30146 65172 1684 54 v 30146 66280 a(\000)p Fs(1)377 b(\026)-805 b Fr(\037)32866 66479 y Fo(j)33353 66280 y Fs(\))p Fr(;)2823 b Fs(0)369 b Fq(\024)g Fr(j)444 b Fq(\024)369 b Fr(h)42597 65732 y Fp(2)p Fo(;)p Fp(1)44224 66280 y Fr(:)0 68730 y Fs(The)434 b(basis)f Fq(f)p 6531 67663 1023 54 v Fr(U)7554 68929 y Fo(j)8041 68730 y Fr(;)p 9057 67663 1048 54 v 655 w(V)10104 68929 y Fo(j)10591 68730 y Fq(g)h Fs(is)g(orthogonal)g(with)g(resp)36 b(ect)433 b(to)h Fr(Q)29534 68929 y Fo(Z)30172 69052 y Fg(0)30688 68730 y Fs(,)g(but)e(not)i(orthonormal.)578 b(By)452 b(\(68\))5948 71762 y Fr(Q)p Fs([)p 7339 70695 1023 54 v Fr(U)8361 71961 y Fo(j)8848 71762 y Fs(])369 b(=)11214 71426 y Fj(e)10959 71762 y Fr(Q)p Fs([)155 b(\026)-805 b Fr(\037)13165 71961 y Fo(j)13652 71762 y Fs(])370 b(=)16815 70863 y(1)p 15896 71457 2488 54 v 15896 72673 a(Im)221 b Fr(\034)18738 71762 y(Q)p Fs([)155 b(\026)-805 b Fr(\037)20944 71961 y Fo(j)21431 71762 y Fs(])369 b(=)24594 70863 y(1)p 23675 71457 V 23675 72673 a(Im)221 b Fr(\034)26295 71762 y(;)2823 b(Q)p Fs([)p 30870 70695 1048 54 v Fr(V)31918 71961 y Fo(j)32405 71762 y Fs(])369 b(=)34770 71426 y Fj(e)34516 71762 y Fr(Q)35767 70686 y Fj(\002)36321 70654 y Fq(p)p 37428 70654 1684 54 v 1108 x(\000)p Fs(1)376 b(\026)-804 b Fr(\037)40148 71961 y Fo(j)40635 70686 y Fj(\003)41557 71762 y Fs(=)43989 70863 y(1)p 43070 71457 2488 54 v 43070 72673 a(Im)222 b Fr(\034)45691 71762 y(:)3636 b Fs(\(79\))p eop %%Page: 28 28 28 27 bop 0 681 a Fm(28)6828 b(MICHAEL)340 b(R.)i(DOUGLAS,)g(BERNARD)e (SHIFFMAN,)i(AND)f(STEVE)f(ZELDITCH)1328 3337 y Fs(T)-108 b(o)443 b(compute)e(det\()p Fr(")11457 3536 y Fo(Z)12095 3659 y Fg(0)12611 3337 y Fq(jS)13884 2855 y Fh(?)13784 3673 y Fo(Z)14422 3796 y Fg(0)14938 3337 y Fs(\),)k(w)-36 b(e)442 b(let)g(\()p Fr(z)21170 3536 y Fp(1)21696 3337 y Fr(;)221 b(:)g(:)g(:)j(;)d(z)25213 3560 y Fo(h)25755 3308 y Fg(2)p Ff(;)p Fg(1)26931 3337 y Fs(\))442 b(b)36 b(e)442 b(normal)g(co)36 b(ordinates)443 b(ab)36 b(out)442 b Fr(z)45325 3536 y Fp(0)46234 3337 y Fq(2)384 b(M)p Fs(,)444 b(and)0 4887 y(w)-36 b(e)434 b(let)g Fq(r)4901 5086 y Fo(j)5387 4887 y Fr(f)576 b Fs(b)36 b(e)433 b(giv)-36 b(en)434 b(b)-36 b(y)18417 6528 y Fq(r)19524 6762 y Fo(@)52 b(=@)g(z)21565 6510 y Ff(j)22055 6528 y Fs(\()p Fr(f)142 b(e)23947 6727 y Fh(L)24649 6528 y Fs(\))369 b(=)f(\()p Fq(r)28517 6727 y Fo(j)29004 6528 y Fr(f)142 b Fs(\))295 b Fq(\012)g Fr(e)32519 6727 y Fh(L)33221 6528 y Fr(;)0 8509 y Fs(for)561 b(1)585 b Fq(\024)h Fr(j)659 b Fq(\024)586 b Fr(h)8533 8027 y Fp(2)p Fo(;)p Fp(1)9790 8509 y Fs(.)959 b(W)-108 b(e)561 b(\014nd)e(it)i(con)-36 b(v)g(enien)g(t)560 b(to)g(use)g(the)g(co)36 b(ordinate)561 b Fr(\034)733 b Fq(2)585 b(E)119 b Fs(,)592 b(although)561 b(it)f(is)h(not)0 10059 y(normal,)434 b(and)f(w)-36 b(e)434 b(use)f(the)g(normalized)h (co)-36 b(v)-72 b(arian)-36 b(t)434 b(deriv)-72 b(ativ)-36 b(e)21246 12242 y Fq(r)22353 12441 y Fp(0)23248 12242 y Fs(:=)369 b(\(Im)221 b Fr(\034)148 b Fs(\))221 b Fq(r)29817 12441 y Fo(\034)30393 12242 y Fr(:)18934 b Fs(\(80\))0 14425 y(No)-36 b(w)434 b(w)-36 b(e)434 b(write)p 14113 14998 1023 54 v 14113 16065 a Fr(U)15135 16264 y Fo(j)15991 16065 y Fs(=)368 b Fr(f)18012 16264 y Fo(j)18499 16065 y Fs(\()p Fr(z)59 b Fs(\))221 b(\012)21334 15517 y Fh(\003)21334 16393 y Fo(z)22161 16065 y Fq(\012)295 b Fr(!)24347 15517 y Fh(\003)24299 16393 y Fo(\034)24874 16065 y Fr(;)p 26758 14998 1048 54 v 1523 w(V)27805 16264 y Fo(j)28661 16065 y Fs(=)368 b Fr(g)30664 16264 y Fo(j)31151 16065 y Fs(\()p Fr(z)59 b Fs(\))221 b(\012)33986 15517 y Fh(\003)33986 16393 y Fo(z)34812 16065 y Fq(\012)296 b Fr(!)36999 15517 y Fh(\003)36951 16393 y Fo(\034)37526 16065 y Fr(;)0 17977 y Fs(where)414 b(the)f(lo)36 b(cal)415 b(frame)f(\012)13577 18176 y Fo(z)14522 17977 y Fs(is)g(normal)f(at)h Fr(z)22371 18176 y Fp(0)22897 17977 y Fs(,)k(and)413 b Fr(!)26995 18176 y Fo(\034)27940 17977 y Fs(=)368 b Fr(dx)255 b Fs(+)f Fr(\034)369 b(dy)48 b Fs(.)572 b(Note)414 b(that)f(the)g(W)-108 b(eil-P)-36 b(etersson)0 19526 y(norm)433 b Fq(j)p Fr(!)4623 19044 y Fh(\003)4575 19855 y Fo(\034)5150 19526 y Fq(j)h Fs(is)g(giv)-36 b(en)434 b(b)-36 b(y)16523 21815 y Fq(j)p Fr(!)17750 21266 y Fh(\003)17702 22143 y Fo(\034)18277 21815 y Fq(j)369 b Fs(=)g Fq(j)p Fr(dx)295 b Fs(+)g Fr(\034)369 b(dy)48 b Fq(j)26443 21266 y Fh(\000)p Fp(1)28070 21815 y Fs(=)31741 20916 y(1)p 29583 21509 4967 54 v 29583 22726 a(\(Im)221 b Fr(\034)148 b Fs(\))33082 22342 y Fp(1)p Fo(=)p Fp(2)35116 21815 y Fr(:)14211 b Fs(\(81\))1328 24527 y(T)-108 b(aking)435 b(in)-36 b(to)433 b(accoun)-36 b(t)434 b(\(79\){\(81\))q(,)f(the)g(Im)222 b Fr(\034)582 b Fs(factors)434 b(cancel)g(out,)g(and)f(w)-36 b(e)434 b(obtain)6605 28326 y(det)o(\()p Fr(")9526 28525 y Fo(Z)10164 28648 y Fg(0)10680 28326 y Fq(jS)11953 27777 y Fh(?)11853 28654 y Fo(Z)12491 28777 y Fg(0)13007 28326 y Fs(\)\))369 b(=)g(det)17929 25656 y Fj(0)17929 28047 y(@)19101 26754 y Fs(Re)221 b Fq(r)21964 26953 y Fo(j)22451 26754 y Fr(f)23092 26953 y Fo(k)24786 26754 y Fs(Re)h Fq(r)27650 26953 y Fo(j)28136 26754 y Fr(g)28759 26953 y Fo(k)19092 29853 y Fs(Im)f Fq(r)21973 30052 y Fo(j)22460 29853 y Fr(f)23101 30052 y Fo(k)24777 29853 y Fs(Im)g Fq(r)27658 30052 y Fo(j)28145 29853 y Fr(g)28768 30052 y Fo(k)29337 25656 y Fj(1)29337 28047 y(A)30499 25603 y(\014)30499 26400 y(\014)30499 27197 y(\014)30499 27994 y(\014)30499 28791 y(\014)30499 29588 y(\014)30942 30440 y Fo(Z)31580 30563 y Fg(0)32318 28326 y Fr(;)1522 b Fs(for)868 b(0)369 b Fq(\024)g Fr(j)r(;)221 b(k)415 b Fq(\024)370 b Fr(h)43408 27777 y Fp(2)p Fo(;)p Fp(1)45034 28326 y Fr(:)1328 32229 y Fs(W)-108 b(e)434 b(no)-36 b(w)434 b(ev)-72 b(aluate)434 b(the)f(en)-36 b(tries)433 b(of)h(the)f(matrix.)579 b(By)434 b(Prop)36 b(osition)435 b(2.2,)f(w)-36 b(e)434 b(ha)-36 b(v)g(e)8201 35040 y Fq(r)9308 35239 y Fo(k)9877 35040 y Fr(f)10518 35239 y Fo(j)11005 35040 y Fs(\()p Fr(Z)95 b Fs(\))369 b(=)14752 33232 y Fj(Z)15490 36239 y Fo(X)p 16610 33972 3406 54 v 16610 35040 a Fq(D)17635 35239 y Fo(j)18121 35040 y Fs(\012)19060 35239 y Fo(z)19498 35362 y Fg(0)20310 35040 y Fq(^)295 b(D)22516 35239 y Fo(k)23084 35040 y Fs(\012)24023 35239 y Fo(z)24554 35040 y Fr(;)1523 b Fq(r)27545 35239 y Fo(k)28113 35040 y Fr(g)28736 35239 y Fo(j)29223 35040 y Fs(\()p Fr(Z)95 b Fs(\))369 b(=)32970 33232 y Fj(Z)33708 36239 y Fo(X)34828 35040 y Fr(i)p 35493 33972 V 221 w Fq(D)36518 35239 y Fo(j)37004 35040 y Fs(\012)37943 35239 y Fo(z)38381 35362 y Fg(0)39193 35040 y Fq(^)295 b(D)41399 35239 y Fo(k)41968 35040 y Fs(\012)42907 35239 y Fo(z)43437 35040 y Fr(;)0 37905 y Fs(and)433 b(hence)10545 39546 y Fq(r)11652 39745 y Fo(j)12139 39546 y Fr(f)12780 39745 y Fo(k)13349 39546 y Fs(\()p 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44208 y Fo(k)36916 44009 y Fr(f)37557 44208 y Fp(0)38451 44009 y Fs(=)g(0)2602 b(for)867 b Fr(k)414 b Fq(\025)370 b Fs(1)p Fr(:)1328 46948 y Fs(By)453 b(\(56\),)434 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)1134 49832 y Fq(r)2241 50031 y Fp(0)2767 49832 y Fs(\()p Fr(f)3914 50031 y Fo(j)4401 49832 y Fs(\))369 b(=)f(\(Im)222 b Fr(\034)148 b Fs(\))221 b Fq(r)11484 50031 y Fo(\034)12059 49832 y Fs(\()p Fr(f)13206 50031 y Fo(j)13693 49832 y Fs(\))369 b(=)15948 48024 y Fj(Z)16686 51032 y Fo(X)17806 49832 y Fq(D)18831 50031 y Fo(j)19318 49832 y Fs(\012)20257 50031 y Fo(z)20695 50154 y Fg(0)21506 49832 y Fq(^)295 b Fs(\012)23626 50031 y Fo(z)24064 50154 y Fg(0)24950 49832 y Fs(=)368 b(0)p Fr(;)1523 b Fq(r)29971 50031 y Fp(0)30497 49832 y Fs(\()p Fr(g)31626 50031 y Fo(j)32112 49832 y Fs(\))369 b(=)g Fq(\000)p Fr(i)36066 48024 y Fj(Z)36804 51032 y Fo(X)37924 49832 y Fs(\012)38863 50031 y Fo(z)39301 50154 y Fg(0)40113 49832 y Fq(^)294 b Fs(\012)42232 50031 y Fo(z)42670 50154 y Fg(0)43556 49832 y Fs(=)368 b(0)p Fr(;)1523 b(j)444 b Fq(\025)369 b Fs(1)p Fr(;)0 52698 y Fs(and)9687 54707 y Fq(r)10794 54906 y Fp(0)11319 54707 y Fs(\()p Fr(f)12466 54906 y Fp(0)12992 54707 y Fs(\))g(=)15247 52899 y Fj(Z)15985 55907 y Fo(X)p 17105 53640 1894 54 v 17105 54707 a Fs(\012)18044 54906 y Fo(z)18482 55029 y Fg(0)19294 54707 y Fq(^)295 b Fs(\012)21414 54906 y Fo(z)21852 55029 y Fg(0)22737 54707 y Fs(=)369 b Fr(i;)1522 b Fq(r)27552 54906 y Fp(0)28078 54707 y Fs(\()p Fr(g)29207 54906 y Fp(0)29732 54707 y Fs(\))369 b(=)31988 52899 y Fj(Z)32726 55907 y Fo(X)p 33845 53640 2338 54 v 33845 54707 a Fr(i)p Fs(\012)35228 54906 y Fo(z)35666 55029 y Fg(0)36478 54707 y Fq(^)295 b Fs(\012)38598 54906 y Fo(z)39036 55029 y Fg(0)39921 54707 y Fs(=)369 b(1)p Fr(:)1328 57302 y Fs(Therefore,)11041 60484 y Fq(j)221 b Fs(det\()p Fr(")14553 60683 y Fo(Z)15191 60806 y Fg(0)15707 60484 y Fq(jS)16980 59936 y Fh(?)16880 60813 y Fo(Z)17518 60936 y Fg(0)18034 60484 y Fs(\)\))p Fq(j)369 b Fs(=)21164 57761 y Fj(\014)21164 58558 y(\014)21164 59355 y(\014)21164 60152 y(\014)21164 60949 y(\014)21164 61746 y(\014)21607 60484 y Fs(det)23635 57814 y Fj(0)23635 60205 y(@)29293 58913 y Fs(0)5602 b Fr(I)24797 62012 y(D)36 b Fs(\(1)p Fr(;)221 b Fq(\000)p Fs(1)p Fr(;)g(:)g(:)g(:)k(;)c Fq(\000)p Fs(1\))1122 b(0)36224 57814 y Fj(1)36224 60205 y(A)37386 57761 y(\014)37386 58558 y(\014)37386 59355 y(\014)37386 60152 y(\014)37386 60949 y(\014)37386 61746 y(\014)38198 60484 y Fs(=)368 b(1)i Fr(:)50967 63913 y Fc(\003)17626 66806 y Fs(5.)651 b Ft(Pr)-30 b(oof)503 b(of)g(Theorem)e(1.4)1328 69131 y Fs(In)456 b(this)f(section)h(w)-36 b(e)456 b(pro)-36 b(v)g(e)456 b(Theorem)f(1.4,)463 b(whic)-36 b(h)455 b(is)h(a)g(com)-36 b(bination)456 b(of)g(an)g(equidistribution) f(the-)0 70681 y(orem)625 b(for)h(radial)g(pro)72 b(jections)626 b(of)g(lattice)g(p)36 b(oin)-36 b(ts)625 b(and)g(an)g(equidistribution) f(theorem)h(for)h(critical)0 72231 y(p)36 b(oin)-36 b(ts.)p eop %%Page: 29 29 29 28 bop 11424 681 a Fm(CRITICAL)440 b(POINTS)j(AND)g(SUPERSYMMETRIC)e (V)-114 b(A)-28 b(CUA,)443 b(I)28 b(I)g(I)10843 b(29)0 3337 y Fs(5.1.)652 b FA(A)470 b(lo)42 b(cal)470 b(v)-83 b(an)470 b(der)g(Corput)g(Theorem.)652 b Fs(W)-108 b(e)409 b(\014rst)f(illustrate)h(the)f(metho)36 b(d)408 b(of)h(pro)36 b(of)410 b(of)f(Theo-)0 4887 y(rem)440 b(1.4)i(b)-36 b(y)440 b(pro)-36 b(viding)441 b(a)f(v)-72 b(an)441 b(der)f(Corput)g(t) -36 b(yp)36 b(e)440 b(asymptotic)h(estimate)g(for)g(the)f(radial)h (distribution)0 6437 y(of)494 b(lattice)f(p)36 b(oin)-36 b(ts)493 b(\(Theorem)g(5.1\).)757 b(The)492 b(estimate)i(has)e(m)-36 b(uc)g(h)492 b(in)h(common)g(with)g(the)f(classical)j(v)-72 b(an)0 7987 y(der)554 b(Corput)g(estimate)h(of)h(Hla)-36 b(wk)-72 b(a,)587 b(Randol)555 b(and)f(others)g(on)h(lattice)g(p)36 b(oin)-36 b(ts)555 b(in)f(dilates)h(of)h(smo)36 b(oth)0 9536 y(con)-36 b(v)g(ex)466 b(sets)g(\(see)f(for)h(example,)475 b([Ra)q(,)465 b(Hl)q(]\),)474 b(and)465 b(w)-36 b(e)465 b(adapt)g(the)g(pro)36 b(of)467 b(of)f(the)f(classical)i(estimate)f(to) 0 11086 y(obtain)434 b(our)f(asymptotic)h(equidistribution)f(theorem.) 1328 12636 y(Let)506 b Fr(Q)493 b Fq(\032)g Fk(R)7739 12154 y Fo(n)8872 12636 y Fs(\()p Fr(n)g Fq(\025)g Fs(2\))506 b(b)36 b(e)507 b(a)f(b)36 b(ounded,)523 b(smo)36 b(oth,)525 b(strictly)507 b(con)-36 b(v)g(ex)507 b(set)f(with)h(0)493 b Fq(2)f Fr(Q)45196 12154 y Fh(\016)45722 12636 y Fs(.)797 b(Let)506 b Fq(j)p Fr(X)104 b Fq(j)51204 12835 y Fo(Q)0 14186 y Fs(denote)433 b(the)g(norm)g(of)i Fr(X)473 b Fq(2)369 b Fk(R)15075 13704 y Fo(n)16135 14186 y Fs(giv)-36 b(en)434 b(b)-36 b(y)18350 16366 y Fr(Q)369 b Fs(=)g Fq(f)p Fr(X)474 b Fq(2)368 b Fk(R)25560 15817 y Fo(n)26556 16366 y Fs(:)h Fq(j)p Fr(X)104 b Fq(j)29207 16565 y Fo(Q)30372 16366 y Fr(<)369 b Fs(1)p Fq(g)221 b Fr(:)16039 b Fs(\(82\))0 18546 y(T)-108 b(o)434 b(measure)f(the)g(equidistribution)g(of)i(pro)72 b(jections)434 b(of)g(lattice)g(p)36 b(oin)-36 b(ts,)434 b(w)-36 b(e)434 b(consider)f(the)g(sums)9269 21471 y Fr(S)10069 21670 y Fo(f)10674 21471 y Fs(\()p Fr(t)p Fs(\))368 b(=)16019 20209 y Fj(X)13904 23087 y Fo(k)24 b Fh(2)p Fi(Z)15635 22835 y Ff(n)16202 23087 y Fh(\\)p Fo(tQ)p Fi(r)p Fh(f)p Fp(0)p Fh(g)20274 21471 y Fr(f)21278 19598 y Fj(\022)23156 20572 y Fr(k)p 22389 21166 2255 54 v 22389 22382 a Fq(j)p Fr(k)45 b Fq(j)23848 22581 y Fo(Q)24776 19598 y Fj(\023)25975 21471 y Fr(;)1523 b Fs(with)867 b Fr(f)511 b Fq(2)368 b Fr(C)34687 20923 y Fh(1)35684 21471 y Fs(\()p Fr(@)72 b(Q)p Fs(\))p Fr(;)654 b(t)369 b(>)g Fs(0)p Fr(:)0 25710 y Fs(W)-108 b(e)509 b(extend)f Fr(f)651 b Fs(to)509 b Fk(R)10533 25228 y Fo(n)11668 25710 y Fs(as)h(a)f(homogeneous)g(function)f(of)i(degree)f (0,)528 b(so)509 b(that)f Fr(f)142 b Fs(\()p Fr(k)45 b Fs(\))497 b(=)g Fr(f)45009 24236 y Fj(\020)46545 25187 y Fo(k)p 45935 25405 1734 54 v 45935 26168 a Fh(j)p Fo(k)24 b Fh(j)46971 26317 y Ff(Q)47802 24236 y Fj(\021)48595 25710 y Fs(.)804 b(Our)0 27680 y(purp)36 b(ose)433 b(is)h(to)f(obtain)h (the)f(follo)-36 b(wing)436 b(asymptotics)e(of)g Fr(S)29046 27879 y Fo(f)29651 27680 y Fs(\()p Fr(t)p Fs(\):)0 30055 y Ft(Theorem)498 b FA(5.1.)14303 32062 y Fr(S)15103 32261 y Fo(f)15708 32062 y Fs(\()p Fr(t)p Fs(\))368 b(=)h Fr(t)19409 31513 y Fo(n)20256 30254 y Fj(Z)20994 33261 y Fo(Q)22011 32062 y Fr(f)363 b(dX)400 b Fs(+)295 b Fr(O)36 b Fs(\()p Fr(t)28483 31479 y Fo(n)p Fh(\000)30229 31117 y Fg(2)p Ff(n)p 29918 31278 1539 40 v 29918 31827 a(n)p Fg(+1)31645 32062 y Fs(\))p Fr(;)959 b(t)369 b Fq(!)g(1)p Fr(:)1328 35365 y Fs(F)-108 b(rom)448 b(this)h(it)f(is)h(simple)g(to)f(obtain)h (asymptotics)g(of)g Fr(S)29197 35564 y Fo(f)29802 35365 y Fs(\()p Fr(t)p Fs(\))f(when)g Fr(f)537 b Fq(2)394 b Fr(C)38627 34883 y Fh(1)39623 35365 y Fs(\()p Fr(@)72 b(Q)p Fs(\))448 b(is)h(extended)f(as)h(a)0 36915 y(homogeneous)434 b(function)f(of)h(an)-36 b(y)434 b(degree)g Fr(\013)442 b Fs(to)434 b Fk(R)24853 36433 y Fo(n)25479 36915 y Fs(:)0 39290 y Ft(Cor)-30 b(ollar)-89 b(y)525 b FA(5.2.)555 b Fl(L)-66 b(et)463 b Fr(f)511 b Fq(2)369 b(C)16036 38808 y Fh(1)17033 39290 y Fs(\()p Fk(R)18498 38808 y Fo(n)19420 39290 y Fk(r)295 b Fq(f)p Fs(0)p Fq(g)p Fs(\))465 b Fl(b)-66 b(e)464 b(homo)-66 b(gene)g(ous)465 b(of)f(de)-66 b(gr)g(e)g(e)463 b Fr(\013)378 b(>)368 b Fs(0)p Fl(,)465 b(and)g(let)17072 41758 y Fr(S)17872 41957 y Fo(f)18477 41758 y Fs(\()p Fr(t)p Fs(\))368 b(=)22751 40496 y Fj(X)21708 43325 y Fo(k)24 b Fh(2)p Fi(Z)23439 43073 y Ff(n)24006 43325 y Fh(\\)p Fo(tQ)25934 41758 y Fr(f)142 b Fs(\()p Fr(k)45 b Fs(\))369 b Fr(;)2878 b(t)369 b(>)g Fs(0)0 45293 y Fl(Then)12967 47300 y Fr(S)13767 47499 y Fo(f)14372 47300 y Fs(\()p Fr(t)p Fs(\))f(=)h Fr(t)18073 46751 y Fo(n)p Fp(+)p Fo(\013)20256 45492 y Fj(Z)20994 48499 y Fo(Q)22011 47300 y Fr(f)363 b(dX)400 b Fs(+)295 b Fr(O)36 b Fs(\()p Fr(t)28483 46717 y Fo(n)p Fh(\000)30229 46355 y Fg(2)p Ff(n)p 29918 46516 V 29918 47064 a(n)p Fg(+1)31589 46717 y Fp(+)p Fo(\013)32981 47300 y Fs(\))p Fr(;)959 b(t)369 b Fq(!)g(1)p Fr(:)0 50529 y Fs(5.1.1.)652 b Fl(Littlewo)-66 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Fo(J)30129 68996 y Fr(r)36 b Fs(\))370 b Fq(!)f Fr(\021)48 b Fs(\()p Fr(r)36 b Fs(\))0 72231 y(b)-36 b(y)433 b(the)h(assumption)f(that)g Fr(\021)416 b Fq(2)369 b Fr(C)17083 71748 y Fh(1)16988 72559 y Fp(0)18079 72231 y Fs(.)p eop %%Page: 30 30 30 29 bop 0 681 a Fm(30)6828 b(MICHAEL)340 b(R.)i(DOUGLAS,)g(BERNARD)e (SHIFFMAN,)i(AND)f(STEVE)f(ZELDITCH)1328 3337 y Fs(W)-108 b(e)434 b(then)f(write)12057 5885 y Fr(S)12857 6084 y Fo(f)13462 5885 y Fs(\()p Fr(t)p Fs(\))1106 b(=)18359 4624 y Fj(X)18169 7452 y Fo(k)24 b Fh(2)p Fi(Z)19900 7200 y Ff(n)20688 5885 y Fr(f)142 b Fs(\()p Fr(k)45 b Fs(\))p Fr(\037)24019 6092 y Fp([0)p Fo(;)p Fp(1])25799 5885 y Fs(\()26438 4987 y Fq(j)p Fr(k)g Fq(j)27897 5186 y Fo(Q)p 26438 5580 2255 54 v 27330 6797 a Fr(t)28825 5885 y Fs(\))802 b(=)g Fr(S)32824 5337 y Fh(0)32747 6214 y Fo(f)33352 5885 y Fs(\()p Fr(t)p Fs(\))295 b(+)g Fr(S)37313 5337 y Fh(00)37236 6214 y Fo(f)37879 5885 y Fs(\()p Fr(t)p Fs(\))p Fr(;)18169 10034 y(S)19046 9486 y Fh(0)18969 10363 y Fo(f)19574 10034 y Fs(\()p Fr(t)p Fs(\))368 b(=)22994 8772 y Fj(X)22805 11601 y Fo(k)24 b Fh(2)p Fi(Z)24536 11349 y Ff(n)25324 10034 y Fr(f)142 b Fs(\()p Fr(k)45 b Fs(\))p Fr(\021)j Fs(\()29169 9136 y Fq(j)p Fr(k)d Fq(j)30628 9335 y Fo(Q)p 29169 9729 V 30060 10945 a Fr(t)31555 10034 y Fs(\))p Fr(;)17266 b Fs(\(83\))18169 14183 y Fr(S)19046 13635 y Fh(00)18969 14511 y Fo(f)19612 14183 y Fs(\()p Fr(t)p Fs(\))802 b(=)23900 12921 y Fj(X)23710 15750 y Fo(k)24 b Fh(2)p Fi(Z)25441 15498 y Ff(n)26229 14183 y Fr(f)142 b Fs(\()p Fr(k)45 b Fs(\)\()p Fr(\037)30066 14390 y Fp([0)p Fo(;)p Fp(1])32141 14183 y Fq(\000)295 b Fr(\021)48 b Fs(\)\()35304 13284 y Fq(j)p Fr(k)d Fq(j)36763 13483 y Fo(Q)p 35304 13877 V 36196 15094 a Fr(t)37691 14183 y Fs(\)\))p Fr(:)10624 b Fs(\(84\))1328 17514 y(W)-108 b(e)548 b(can)g(assume)g(without)g(loss)h(of)g(generalit)-36 b(y)549 b(that)e Fr(f)706 b Fq(\025)564 b Fs(0.)922 b(W)-108 b(e)548 b(b)36 b(egin)548 b(with)g(the)f(\014rst)h(sum)f(in)0 19064 y Fr(S)877 18582 y Fh(0)800 19409 y Fo(f)1405 19064 y Fs(\()p Fr(t)p Fs(\):)0 21222 y Ft(Lemma)497 b FA(5.3.)14097 23141 y Fr(S)14974 22592 y Fh(0)14897 23469 y Fo(f)15502 23141 y Fs(\()p Fr(t)p Fs(\))368 b(=)g Fr(t)19202 22592 y Fo(n)20050 21333 y Fj(Z)20788 24340 y Fi(R)21428 24088 y Ff(n)22271 23141 y Fr(f)142 b Fs(\()p Fr(X)104 b Fs(\))p Fr(\021)48 b Fs(\()p Fq(j)p Fr(X)104 b Fq(j)28366 23340 y Fo(Q)29162 23141 y Fs(\))p Fr(dX)399 b Fs(+)295 b Fr(O)36 b Fs(\(log)241 b Fr(t)p Fs(\))p Fr(:)0 25981 y Fl(Pr)-66 b(o)g(of.)552 b Fs(W)-108 b(e)434 b(write)g(the)f(sum)g(as)17244 29067 y Fr(S)18121 28519 y Fh(0)18044 29396 y Fo(f)18649 29067 y Fs(\()p Fr(t)p Fs(\))368 b(=)22369 27407 y Fh(1)21880 27805 y Fj(X)22022 30605 y Fo(j)51 b Fp(=0)24210 27805 y Fj(X)24020 30634 y Fo(k)24 b Fh(2)p Fi(Z)25751 30382 y Ff(n)26539 29067 y Fr(f)142 b Fs(\()p Fr(k)45 b Fs(\))p Fr(\032)p Fs(\()30365 28169 y(2)31015 27687 y Fo(j)31502 28169 y Fq(j)p Fr(k)g Fq(j)32961 28368 y Fo(Q)p 30365 28762 3392 54 v 31825 29978 a Fr(t)33889 29067 y Fs(\))p Fr(:)1328 32873 y Fs(W)-108 b(e)504 b(further)f(break)h(up)e(the)i(dy) -36 b(adic)503 b(sum)h(in)-36 b(to)25705 31877 y Fj(P)27107 32227 y Fo(J)83 b Fp(\()p Fo(t)p Fp(\))27107 33260 y Fo(j)51 b Fp(=0)29053 32873 y Fs(+)30286 31877 y Fj(P)31688 32227 y Fh(1)31688 33260 y Fo(j)g Fp(=)p Fo(J)83 b Fp(\()p Fo(t)p Fp(\)+1)36281 32873 y Fs(with)504 b Fr(J)123 b Fs(\()p Fr(t)p Fs(\))503 b(to)h(b)36 b(e)503 b(determined)0 34515 y(later.)579 b(W)-108 b(e)433 b(\014rst)g(consider)17844 37186 y Fr(S)18721 36638 y Fh(0)18644 37515 y Fp(1)19538 37186 y Fs(:=)21405 35422 y Fo(J)83 b Fp(\()p Fo(t)p Fp(\))21280 35924 y Fj(X)21423 38724 y Fo(j)51 b Fp(=0)23610 35924 y Fj(X)23420 38753 y Fo(k)24 b Fh(2)p Fi(Z)25151 38501 y Ff(n)25940 37186 y Fr(f)142 b Fs(\()p Fr(k)45 b Fs(\))p Fr(\032)p Fs(\()29766 36288 y(2)30416 35806 y Fo(j)30902 36288 y Fq(j)p Fr(k)g Fq(j)32361 36487 y Fo(Q)p 29766 36881 V 31226 38097 a Fr(t)33289 37186 y Fs(\))p Fr(:)0 40478 y Fs(The)437 b(function)g Fr(f)142 b Fs(\()p Fr(X)104 b Fs(\))p Fr(\032)p Fs(\(2)12580 39996 y Fo(j)13067 40478 y Fq(j)p Fr(X)g Fq(j)14988 40677 y Fo(Q)15785 40478 y Fs(\))374 b Fq(2)h Fr(C)18952 39996 y Fh(1)18857 40806 y Fp(0)19948 40478 y Fs(\()p Fk(R)21413 39996 y Fo(n)22040 40478 y Fs(\))437 b(when)f Fr(f)579 b Fs(is)438 b(homogeneous)f(of)h(degree)f(0)g(and)g(smo)36 b(oth)437 b(on)0 42028 y Fr(@)72 b(Q)p Fs(.)578 b(Hence)434 b(w)-36 b(e)434 b(ma)-36 b(y)434 b(apply)g(the)f(P)-36 b(oisson)434 b(summation)f(form)-36 b(ula)434 b(to)g(the)f Fr(k)478 b Fs(sum)433 b(to)h(obtain)13226 45500 y Fr(S)14103 44952 y Fh(0)14026 45828 y Fp(1)14921 45500 y Fs(=)16426 43736 y Fo(J)83 b Fp(\()p Fo(t)p Fp(\))16301 44238 y Fj(X)16444 47037 y Fo(j)51 b Fp(=0)18795 44238 y Fj(X)18441 47062 y Fo(N)94 b Fh(2)p Fi(Z)20499 46810 y Ff(n)21288 43692 y Fj(Z)22026 46699 y Fi(R)22666 46447 y Ff(n)23509 45500 y Fr(e)24112 44952 y Fh(\000)p Fo(i)p Fh(h)p Fo(X)17 b(;N)94 b Fh(i)27844 45500 y Fr(f)142 b Fs(\()p Fr(X)104 b Fs(\))p Fr(\032)p Fs(\()32132 44601 y(2)32782 44119 y Fo(j)33269 44601 y Fq(j)p Fr(X)g Fq(j)35190 44800 y Fo(Q)p 32132 45194 3855 54 v 33825 46411 a Fr(t)36120 45500 y Fs(\))p Fr(dX)32 b(:)1328 48918 y Fs(The)434 b(terms)f(with)h Fr(N)507 b Fs(=)369 b(0)434 b(sum)f(up)f(to)2503 52553 y Fr(t)2973 52004 y Fo(n)3820 50745 y Fj(Z)4558 53752 y Fi(R)5198 53500 y Ff(n)6041 52553 y Fr(f)142 b Fs(\()p Fr(X)104 b Fs(\))9240 49829 y Fj(8)9240 51025 y(<)9240 53416 y(:)10546 50789 y Fo(J)83 b Fp(\()p Fo(t)p Fp(\))10421 51291 y Fj(X)10564 54090 y Fo(j)51 b Fp(=0)12561 52553 y Fr(\032)p Fs(\(2)14388 52004 y Fo(j)14875 52553 y Fq(j)p Fr(X)104 b Fq(j)16796 52752 y Fo(Q)17593 52553 y Fs(\))18099 49829 y Fj(9)18099 51025 y(=)18099 53416 y(;)19501 52553 y Fr(dX)1211 b Fs(=)1107 b Fr(t)25056 52004 y Fo(n)25903 50745 y Fj(Z)26641 53752 y Fi(R)27281 53500 y Ff(n)28124 52553 y Fr(f)142 b Fs(\()p Fr(X)104 b Fs(\))31323 51477 y Fj(\010)32099 52553 y Fr(\021)48 b Fs(\()p Fq(j)p Fr(X)104 b Fq(j)35216 52752 y Fo(Q)36011 52553 y Fs(\))295 b Fq(\000)h Fr(\021)48 b Fs(\(2)39987 52004 y Fo(J)83 b Fp(\()p Fo(t)p Fp(\)+1)42913 52553 y Fq(j)p Fr(X)104 b Fq(j)44834 52752 y Fo(Q)45630 52553 y Fs(\)\))46642 51477 y Fj(\011)47638 52553 y Fr(dX)22467 56918 y Fs(=)1107 b Fr(t)25056 56369 y Fo(n)25903 55110 y Fj(Z)26641 58117 y Fi(R)27281 57865 y Ff(n)28124 56918 y Fr(f)142 b Fs(\()p Fr(X)104 b Fs(\))p Fr(\021)48 b Fs(\()p Fq(j)p Fr(X)104 b Fq(j)34219 57117 y Fo(Q)35015 56918 y Fs(\))p Fr(dX)400 b Fs(+)295 b Fr(O)36 b Fs(\()p Fr(t)40989 56369 y Fo(n)41614 56918 y Fs(2)42264 56369 y Fh(\000)p Fo(nJ)83 b Fp(\()p Fo(t)p Fp(\))45292 56918 y Fs(\))p Fr(;)0 59916 y Fs(where)595 b(the)h(last)g(estimate)f(is)h(a) g(consequence)g(of)g(the)f(fact)h(that)f Fr(\021)48 b Fs(\(2)36137 59434 y Fo(J)83 b Fp(\()p Fo(t)p Fp(\)+1)39063 59916 y Fq(j)p Fr(X)104 b Fq(j)40984 60115 y Fo(Q)41781 59916 y Fs(\))595 b(is)h(supp)36 b(orted)594 b(on)0 61545 y(2)650 61063 y Fh(\000)p Fo(J)83 b Fp(\()p Fo(t)p Fp(\))3106 61545 y Fr(Q)p Fs(.)1328 63094 y(T)-108 b(o)308 b(estimate)g(the)g (remaining)f(terms)h(in)f(the)h(sum)f Fr(S)26764 62612 y Fh(0)26687 63423 y Fp(1)27213 63094 y Fs(,)333 b(w)-36 b(e)308 b(mak)-36 b(e)308 b(the)f(c)-36 b(hange)308 b(of)g(v)-72 b(ariables)309 b Fr(Y)658 b Fs(=)368 b(2)49317 62612 y Fo(j)49804 63094 y Fr(X)l(=t)0 64644 y Fs(in)433 b(the)g(in)-36 b(tegral)434 b(to)g(obtain)15879 67364 y(2)16529 66815 y Fh(\000)p Fo(nj)18319 67364 y Fr(t)18789 66815 y Fo(n)19636 65556 y Fj(Z)20374 68563 y Fi(R)21014 68311 y Ff(n)21857 67364 y Fr(f)142 b Fs(\()p Fr(Y)288 b Fs(\))p Fr(\032)p Fs(\()p Fq(j)p Fr(Y)g Fq(j)p Fs(\))p Fr(e)28762 66815 y Fh(\000)p Fo(i)p Fp(2)30284 66503 y Fe(\000)p Ff(j)31358 66815 y Fo(t)p Fh(h)p Fo(Y)52 b(;N)94 b Fh(i)34185 67364 y Fr(d)72 b(Y)g(:)0 70138 y Fs(Since)433 b(the)g(in)-36 b(tegrand)433 b(is)h(smo)36 b(oth,)434 b(this)f(term)g(has)h(the)f(upp) 36 b(er)432 b(b)36 b(ound)16064 72231 y Fr(c)221 b Fs(2)17495 71682 y Fh(\000)p Fo(nj)19285 72231 y Fr(t)19755 71682 y Fo(n)20381 72231 y Fs(\(1)295 b(+)g(2)23789 71682 y Fh(\000)p Fo(j)25008 72231 y Fq(j)p Fr(N)139 b Fq(j)p Fr(t)p Fs(\))27903 71682 y Fh(\000)p Fo(K)29545 72231 y Fr(;)1329 b Fq(8)p Fr(K)464 b(>)368 b Fs(0)p Fr(:)p eop %%Page: 31 31 31 30 bop 11424 681 a Fm(CRITICAL)440 b(POINTS)j(AND)g(SUPERSYMMETRIC)e (V)-114 b(A)-28 b(CUA,)443 b(I)28 b(I)g(I)10843 b(31)0 3337 y Fs(\(Again,)527 b(w)-36 b(e)508 b(let)f Fr(c)h Fs(denote)f(a)h(constan)-36 b(t;)544 b Fr(c)507 b Fs(dep)36 b(ends)506 b(on)i Fr(f)649 b Fs(and)507 b Fr(K)95 b Fs(,)527 b(but)506 b(is)i(indep)36 b(enden)-36 b(t)506 b(of)i Fr(j)r(;)221 b(t;)g(N)139 b Fs(.\))0 4887 y(The)434 b(sum)f(o)-36 b(v)g(er)434 b Fr(N)507 b Fq(6)p Fs(=)369 b(0)433 b(is)h(then)f(b)36 b(ounded)432 b(b)-36 b(y)4379 8031 y Fr(c)221 b(t)5630 7482 y Fo(n)6934 6769 y Fj(X)6477 9647 y Fo(j)51 b Fh(\024)p Fo(J)83 b Fp(\()p Fo(t)p Fp(\))9531 8031 y Fs(2)10181 7482 y Fh(\000)p Fo(nj)12255 6769 y Fj(X)12192 9598 y Fo(N)94 b Fh(6)p Fp(=0)14236 8031 y Fs(\(1)295 b(+)g(2)17644 7482 y Fh(\000)p Fo(j)18863 8031 y Fq(j)p Fr(N)139 b Fq(j)p Fr(t)p Fs(\))21758 7482 y Fh(\000)p Fo(K)24507 8031 y Fq(\030)1107 b Fr(t)27117 7482 y Fo(n)28421 6769 y Fj(X)27964 9647 y Fo(j)51 b Fh(\024)p Fo(J)83 b Fp(\()p Fo(t)p Fp(\))31018 8031 y Fs(2)31668 7482 y Fh(\000)p Fo(nj)33679 6223 y Fj(Z)35008 6573 y Fh(1)34417 9230 y Fp(0)36004 8031 y Fs(\(1)295 b(+)g(2)39412 7482 y Fh(\000)p Fo(j)40631 8031 y Fr(r)36 b(t)p Fs(\))42229 7482 y Fh(\000)p Fo(K)43872 8031 y Fr(r)44494 7482 y Fo(n)p Fh(\000)p Fp(1)46323 8031 y Fr(dr)24518 12338 y Fs(=)27104 11076 y Fj(X)26647 13954 y Fo(j)51 b Fh(\024)p Fo(J)83 b Fp(\()p Fo(t)p Fp(\))29701 10530 y Fj(Z)31029 10881 y Fh(1)30439 13537 y Fp(0)32025 12338 y Fs(\(1)296 b(+)e Fr(s)p Fs(\))35902 11790 y Fh(\000)p Fo(K)37545 12338 y Fr(s)38158 11790 y Fo(n)p Fh(\000)p Fp(1)39986 12338 y Fr(ds)802 b Fs(=)h Fr(c)221 b(J)123 b Fs(\()p Fr(t)p Fs(\))p Fr(:)0 16277 y Fs(Therefore)10692 18589 y Fr(S)11569 18041 y Fh(0)11492 18918 y Fp(1)12387 18589 y Fs(=)369 b Fr(t)14238 18041 y Fo(n)15085 16781 y Fj(Z)15823 19789 y Fi(R)16463 19536 y Ff(n)17306 18589 y Fr(f)142 b Fs(\()p Fr(X)104 b Fs(\))p Fr(\021)48 b Fs(\()p Fq(j)p Fr(X)104 b Fq(j)23401 18788 y Fo(Q)24197 18589 y Fs(\))p Fr(dX)400 b Fs(+)294 b Fr(O)36 b Fs(\()p Fr(t)30170 18041 y Fo(n)30796 18589 y Fs(2)31446 18041 y Fh(\000)p Fo(nJ)83 b Fp(\()p Fo(t)p Fp(\))34474 18589 y Fs(\))295 b(+)f Fr(O)36 b Fs(\()p Fr(J)123 b Fs(\()p Fr(t)p Fs(\)\))p Fr(:)1328 21560 y Fs(Recall)435 b(that)e Fr(S)8943 21078 y Fh(0)8866 21906 y Fo(f)9471 21560 y Fs(\()p Fr(t)p Fs(\))368 b(=)h Fr(S)13579 21078 y Fh(0)13502 21888 y Fp(1)14323 21560 y Fs(+)295 b Fr(S)16507 21078 y Fh(0)16430 21888 y Fp(2)16956 21560 y Fs(,)433 b(where)16067 25299 y Fr(S)16944 24750 y Fh(0)16867 25627 y Fp(2)17762 25299 y Fs(=)20689 23638 y Fh(1)20200 24037 y Fj(X)19142 26915 y Fo(j)51 b Fp(=)p Fo(J)83 b Fp(\()p Fo(t)p Fp(\)+1)23588 24037 y Fj(X)23398 26866 y Fo(k)24 b Fh(2)p Fi(Z)25129 26614 y Ff(n)25917 25299 y Fr(f)142 b Fs(\()28106 24400 y Fr(k)p 27339 24993 2255 54 v 27339 26210 a Fq(j)p Fr(k)45 b Fq(j)28798 26409 y Fo(Q)29726 25299 y Fs(\))p Fr(\032)p Fs(\()31542 24400 y(2)32192 23918 y Fo(j)32679 24400 y Fq(j)p Fr(k)g Fq(j)34138 24599 y Fo(Q)p 31542 24993 3392 54 v 33002 26210 a Fr(t)35066 25299 y Fs(\))p Fr(:)0 29223 y Fs(Since)14449 30241 y Fh(1)13960 30640 y Fj(X)12903 33518 y Fo(j)51 b Fp(=)p Fo(J)83 b Fp(\()p Fo(t)p Fp(\)+1)17158 31902 y Fr(\032)p Fs(\()18468 31003 y(2)19118 30521 y Fo(j)19605 31003 y Fq(j)p Fr(k)45 b Fq(j)21064 31202 y Fo(Q)p 18468 31596 V 19929 32813 a Fr(t)21992 31902 y Fs(\))369 b(=)g Fr(\021)25159 30029 y Fj(\022)26269 31003 y Fs(2)26919 30521 y Fo(J)83 b Fp(\()p Fo(t)p Fp(\))28644 31003 y Fq(j)p Fr(k)45 b Fq(j)30103 31202 y Fo(Q)p 26269 31596 4629 54 v 28349 32813 a Fr(t)31031 30029 y Fj(\023)32378 31902 y Fq(\024)369 b Fr(\037)34595 32204 y Fo(t)p Fp(2)35405 31942 y Fe(\000)p Ff(J)63 b Fg(\()p Ff(t)p Fg(\))37572 32204 y Fo(Q)38736 31902 y Fr(;)0 35794 y Fs(the)386 b(remainder)g Fr(S)9165 35312 y Fh(0)9088 36122 y Fp(2)10001 35794 y Fs(is)h(b)36 b(ounded)385 b(b)-36 b(y)387 b(the)f(total)i(n)-36 b(um)g(b)36 b(er)385 b(of)i(lattice)h(p)36 b(oin)-36 b(ts)386 b(in)h(the)f(shell)h Fq(j)p Fr(k)45 b Fq(j)45496 35993 y Fo(Q)46661 35794 y Fq(\024)369 b Fs(2)48713 35312 y Fh(\000)p Fo(J)83 b Fp(\()p Fo(t)p Fp(\))51169 35794 y Fr(t)p Fs(,)0 37422 y(hence)433 b(is)h(of)g(order)f Fr(t)10268 36940 y Fo(n)10894 37422 y Fs(2)11544 36940 y Fh(\000)p Fo(nJ)83 b Fp(\()p Fo(t)p Fp(\))14571 37422 y Fs(.)579 b(It)433 b(follo)-36 b(ws)436 b(that)9912 40276 y Fr(S)10789 39728 y Fh(0)10712 40605 y Fo(f)11317 40276 y Fs(\()p Fr(t)p Fs(\))368 b(=)h Fr(t)15018 39728 y Fo(n)15865 38468 y Fj(Z)16603 41476 y Fi(R)17243 41223 y Ff(n)18086 40276 y Fr(f)142 b Fs(\()p Fr(X)104 b Fs(\))p Fr(\021)48 b Fs(\()p Fq(j)p Fr(X)104 b Fq(j)24181 40475 y Fo(Q)24977 40276 y Fs(\))p Fr(dX)400 b Fs(+)295 b Fr(O)36 b Fs(\()p Fr(t)30951 39728 y Fo(n)31577 40276 y Fs(2)32227 39728 y Fh(\000)p Fo(nJ)83 b Fp(\()p Fo(t)p Fp(\))35254 40276 y Fs(\))295 b(+)g Fr(O)36 b Fs(\()p Fr(J)123 b Fs(\()p Fr(t)p Fs(\)\))p Fr(:)7599 b Fs(\(85\))0 43518 y(T)-108 b(o)432 b(balance)f(the)g(terms,)g(w)-36 b(e)432 b(c)-36 b(ho)36 b(ose)432 b Fr(J)123 b Fs(\()p Fr(t)p Fs(\))368 b(=)g(log)24613 43832 y Fp(2)25361 43518 y Fr(t)p Fs(,)431 b(and)g(then)f(the)h(last)h(t)-36 b(w)g(o)431 b(terms)g(of)525 b(\(85\))431 b(ha)-36 b(v)g(e)432 b(the)0 45068 y(form)17248 47011 y Fr(O)36 b Fs(\()p Fr(t)19254 46463 y Fo(n)19880 47011 y Fr(t)20350 46463 y Fh(\000)p Fo(n)21708 47011 y Fs(\))295 b(+)f Fr(O)36 b Fs(\(log)241 b Fr(t)p Fs(\))368 b(=)h Fr(O)36 b Fs(\(log)240 b Fr(t)p Fs(\))p Fr(:)50967 49225 y Fc(\003)0 52337 y Fs(5.1.2.)652 b Fl(Stationary)533 b(phase.)651 b Fs(Theorem)509 b(5.1)i(is)f(an)f (immediate)h(consequence)f(of)h(Lemma)f(5.3)i(and)e(the)0 53886 y(follo)-36 b(wing)436 b(assymptotics)e(of)g(the)f(second)h(sum)f Fr(S)24759 53404 y Fh(00)24682 54232 y Fo(f)25325 53886 y Fs(\()p Fr(t)p 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Ff(n)9013 12009 y Fi(r)p Fo(tT)10633 12120 y Ff(")11299 8569 y Fj(\024)12000 10442 y Fr(g)12892 8569 y Fj(\022)14003 9544 y Fr(k)p 14003 10137 722 54 v 14129 11354 a(t)14856 8569 y Fj(\023)16129 10442 y Fq(\000)296 b Fr(g)18081 10641 y Fo(")18792 8569 y Fj(\022)19902 9544 y Fr(k)p 19902 10137 V 20028 11354 a(t)20756 8569 y Fj(\023\025)22435 8118 y(\014)22435 8915 y(\014)22435 9712 y(\014)22435 10509 y(\014)22435 11306 y(\014)23247 10442 y Fq(\024)26891 9180 y Fj(X)24649 12059 y Fo(k)24 b(=t)p Fh(2)p Fp(Supp)156 b Fo(g)32 b Fi(r)p Fo(T)30608 12170 y Ff(")31274 10442 y Fr(")874 b Fs(sup)32104 11586 y Fh(j)p Fo(X)69 b Fh(j)33469 11735 y Ff(Q)34167 11586 y Fo(>)p Fp(1)35590 10442 y Fq(j)p Fr(dg)48 b Fs(\()p Fr(X)104 b Fs(\))p Fq(j)369 b Fs(=)g Fr(O)36 b Fs(\()p Fr("t)44235 9894 y Fo(n)44860 10442 y Fs(\))p Fr(;)0 14831 y Fs(whic)-36 b(h)433 b(v)-36 b(eri\014es)434 b(\(86\).)1328 16381 y(The)g(P)-36 b(oisson)434 b(summation)f(form)-36 b(ula)434 b(then)f(giv)-36 b(es)9438 18195 y Fj(X)9249 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Fr("tN)139 b Fq(j)p Fs(\))34642 70253 y Fh(\000)p Fo(K)36283 70801 y Fr(:)13044 b Fs(\(88\))p eop %%Page: 33 33 33 32 bop 11424 681 a Fm(CRITICAL)440 b(POINTS)j(AND)g(SUPERSYMMETRIC)e (V)-114 b(A)-28 b(CUA,)443 b(I)28 b(I)g(I)10843 b(33)0 3337 y Fs(The)434 b(sum)f(\(88\))h(can)f(b)36 b(e)433 b(replaced)h(b)-36 b(y)433 b(the)g(in)-36 b(tegral)1602 6224 y Fr(c)221 b(t)2853 5675 y Fo(n)3700 4416 y Fj(Z)4438 7423 y Fi(R)5078 7171 y Ff(n)5700 6224 y Fs(\(1)295 b(+)g Fq(j)p Fr(tN)139 b Fq(j)p Fs(\))11353 5675 y Fh(\000)12218 5228 y Fg(\()p Ff(n)p Fg(+1\))p 12216 5474 2178 40 v 13102 6023 a(2)14582 6224 y Fs(\(1)295 b(+)g Fq(j)p Fr("tN)139 b Fq(j)p Fs(\))20844 5675 y Fh(\000)p Fo(K)22707 6224 y Fr(dN)1256 b Fs(=)1118 b Fr(c)221 b(t)29062 5675 y Fo(n)29909 4416 y Fj(Z)31237 4766 y Fh(1)30647 7423 y Fp(0)32234 6224 y Fs(\(1)295 b(+)g Fr(tr)36 b Fs(\))36590 5675 y Fh(\000)37455 5228 y Fg(\()p Ff(n)p Fg(+1\))p 37454 5474 V 38340 6023 a(2)39820 6224 y Fs(\(1)296 b(+)e Fr("tr)36 b Fs(\))44785 5675 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70614 y Fr(t)13705 70066 y Fo(\013)14364 70614 y Fr(d\026)p Fs(\()p Fr(t)p Fs(\))368 b(=)19532 69352 y Fj(X)19053 72171 y Fo(j)51 b Fp(:)p Fo(t)20086 72306 y Ff(j)20520 72171 y Fh(\024)p Fo(T)22151 67891 y Fj(8)22151 69087 y(<)22151 71478 y(:)24379 69352 y(X)23332 72231 y Fo(k)24 b Fp(:)p Fh(j)p Fo(k)g Fh(j)25143 72380 y Ff(Q)25840 72231 y Fp(=)p Fo(t)26912 72366 y Ff(j)27567 70614 y Fr(f)28571 68741 y Fj(\022)30449 69716 y Fr(k)p 29682 70309 V 29682 71526 a Fq(j)p Fr(k)45 b Fq(j)31141 71725 y Fo(Q)32069 68741 y Fj(\023)33047 67891 y(9)33047 69087 y(=)33047 71478 y(;)34449 70614 y Fr(t)34919 70066 y Fo(\013)34919 70943 y(j)35948 70614 y Fs(=)368 b Fr(S)38128 70813 y Fo(f)38733 70614 y Fs(\()p Fr(T)181 b Fs(\))p Fr(:)p eop %%Page: 34 34 34 33 bop 0 681 a Fm(34)6828 b(MICHAEL)340 b(R.)i(DOUGLAS,)g(BERNARD)e (SHIFFMAN,)i(AND)f(STEVE)f(ZELDITCH)0 3337 y Fs(In)-36 b(tegrating)452 b(\(90\))434 b(b)-36 b(y)433 b(parts)g(and)g(applying) 452 b(\(89\))q(,)433 b(w)-36 b(e)434 b(get)1882 6928 y Fr(S)2682 7127 y Fo(f)3287 6928 y Fs(\()p Fr(T)181 b Fs(\))1107 b(=)f Fr(T)9410 6380 y Fo(\013)10070 6928 y Fr(\026)p Fs(\()p Fr(T)181 b Fs(\))294 b Fq(\000)i Fr(\013)15488 5120 y Fj(Z)16816 5471 y Fo(T)16226 8128 y Fp(0)17771 6928 y Fr(t)18241 6380 y Fo(\013)p Fh(\000)p Fp(1)20103 6928 y Fr(\026)p Fs(\()p Fr(t)p Fs(\))p Fr(dt)6349 10688 y Fs(=)1106 b Fr(T)9410 10140 y Fo(\013)10291 8815 y Fj(\024)10992 10688 y Fr(T)11935 10140 y Fo(n)12783 8880 y Fj(Z)13521 11888 y Fo(Q)14538 10688 y Fr(f)15179 10887 y Fp(0)15926 10688 y Fr(dX)400 b Fs(+)295 b Fr(O)36 b Fs(\()p Fr(T)21867 10106 y Fo(n)p Fh(\000)23613 9744 y Fg(2)p Ff(n)p 23302 9905 1539 40 v 23302 10453 a(n)p Fg(+1)25029 10688 y Fs(\))25535 8815 y Fj(\025)26531 10688 y Fq(\000)295 b Fr(\013)28916 8880 y Fj(Z)30245 9231 y Fo(T)29654 11888 y Fp(0)31200 10688 y Fr(t)31670 10140 y Fo(\013)p Fh(\000)p Fp(1)33753 8815 y Fj(\024)34454 10688 y Fr(t)34924 10140 y Fo(n)35771 8880 y Fj(Z)36509 11888 y Fo(Q)37526 10688 y Fr(f)38167 10887 y Fp(0)38915 10688 y Fr(dX)399 b Fs(+)295 b Fr(O)36 b Fs(\()p Fr(t)44382 10106 y Fo(n)p Fh(\000)46128 9744 y Fg(2)p Ff(n)p 45818 9905 V 45818 10453 a(n)p Fg(+1)47544 10688 y Fs(\))48050 8815 y Fj(\025)48972 10688 y Fr(dt)6349 14484 y Fs(=)1106 b Fr(T)9410 13936 y Fo(n)p Fp(+)p Fo(\013)11594 12611 y Fj(\024)12295 12676 y(Z)13033 15683 y Fo(Q)14050 14484 y Fr(f)14691 14683 y Fp(0)15438 14484 y Fr(dX)17297 12611 y Fj(\025)19572 13585 y Fr(n)p 18353 14179 3215 54 v 18353 15395 a(\013)304 b Fs(+)295 b Fr(n)21996 14484 y Fs(+)f Fr(O)36 b Fs(\()p Fr(T)25781 13902 y Fo(n)p Fh(\000)27527 13539 y Fg(2)p Ff(n)p 27217 13700 1539 40 v 27217 14249 a(n)p Fg(+1)28888 13902 y Fp(+)p Fo(\013)30279 14484 y Fs(\))6349 18162 y(=)1106 b Fr(T)9410 17613 y Fo(n)p Fp(+)p Fo(\013)11594 16354 y Fj(Z)12332 19361 y Fo(Q)13349 18162 y Fr(f)363 b(dX)400 b Fs(+)295 b Fr(O)36 b Fs(\()p Fr(T)20294 17579 y Fo(n)p Fh(\000)22040 17217 y Fg(2)p Ff(n)p 21730 17378 V 21730 17926 a(n)p Fg(+1)23401 17579 y Fp(+)p Fo(\013)24792 18162 y Fs(\))221 b Fr(:)50967 21626 y Fc(\003)0 25037 y Fs(5.2.)652 b FA(V)-125 b(an)524 b(der)g(Corput)h(for)e(critical)i(p)42 b(oin)-42 b(ts.)651 b Fs(W)-108 b(e)456 b(pro)-36 b(v)g(e)455 b(Theorem)h(1.4)g(b)-36 b(y)455 b(follo)-36 b(wing)458 b(the)c(argu-)0 26586 y(men)-36 b(ts)540 b(of)i(the)e(pro)36 b(ofs)542 b(of)f(Theorem)g(5.1)h (and)e(Corollary)j(5.2)f(with)f(hardly)g(an)-36 b(y)541 b(c)-36 b(hanges.)900 b(W)-108 b(e)541 b(\014rst)0 28136 y(assume)421 b(that)g Fr( )468 b Fs(is)421 b(homogeneous)g(of)h(order)f (0)g(in)g Fq(S)100 b Fs(.)574 b(W)-108 b(e)421 b(let)h Fr(K)32309 28335 y Fo( )33379 28136 y Fs(=)369 b Fr(\032)p Fs(\(Supp)219 b Fr( )48 b Fs(\))368 b Fq(\032)i(C)77 b Fs(,)424 b(a)e(compact)f(set.)1328 29686 y(T)-108 b(o)446 b(b)36 b(egin,)449 b(w)-36 b(e)446 b(recall)h(that)e(if)h Fr(W)627 b Fs(has)445 b(critical)i(p)36 b(oin)-36 b(ts,)449 b(then)444 b Fr(W)626 b Fs(is)446 b(in)g(the)f(`ligh)-36 b(t)446 b(cone')g Fr(Q)p Fs([)p Fr(W)181 b Fs(])390 b Fr(>)f Fs(0.)0 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b Fr(Q)295 b Fq(\\)g(S)22469 44140 y Fo( )23172 43941 y Fr(;)2823 b Fq(S)27160 44140 y Fo( )28231 43941 y Fs(=)29612 41271 y Fj(0)29612 43662 y(@)31318 42679 y([)30774 45498 y Fo(\034)102 b Fh(2)p Fo(K)32714 45654 y Ff( )33559 43941 y Fq(S)34363 44140 y Fo(\034)34939 41271 y Fj(1)34939 43662 y(A)36323 43941 y Fr(;)0 48118 y Fs(whic)-36 b(h)433 b(is)h(a)g(compact)g(subset)e(of)j Fq(S)100 b Fs(.)1328 49668 y(W)-108 b(e)434 b(let)g Fr(f)142 b Fs(\()p Fr(W)181 b Fs(\))368 b(=)g Fq(h)p Fr(C)11847 49867 y Fo(W)12920 49668 y Fr(;)221 b( )48 b Fq(i)p Fs(,)434 b(whic)-36 b(h)433 b(is)h(a)g(smo)36 b(oth)433 b(function)h(supp)36 b(orted)432 b(in)h Fq(S)39945 49867 y Fo( )40648 49668 y Fs(.)578 b(Then)15512 52631 y Fq(N)16602 52830 y Fo( )17304 52631 y Fs(\()p Fr(L)p Fs(\))368 b(=)h Fr(S)21750 52830 y Fo(f)22355 52631 y Fs(\()p Fr(L)p Fs(\))f(=)28736 51369 y Fj(X)26001 54413 y Fo(k)24 b Fh(2)p Fi(Z)27732 54160 y Ff(n)28299 54413 y Fh(\\)28926 53662 y(p)p 29710 53662 640 40 v 751 x Fo(L)157 b(Q)p Fi(r)p Fh(f)p Fp(0)p Fh(g)33611 52631 y Fr(f)142 b Fs(\()p Fr(k)45 b Fs(\))p Fr(;)0 57194 y Fs(as)645 b(b)36 b(efore.)1214 b(No)-36 b(w)645 b(w)-36 b(e)646 b(follo)-36 b(w)647 b(the)d(previous)i(pro)36 b(of,)699 b(with)645 b Fr(t)728 b Fs(=)34092 56069 y Fq(p)p 35199 56069 885 54 v 1125 x Fr(L)p Fs(.)1213 b(Our)644 b(\014rst)g(mo)36 b(di\014cation)646 b(is)0 58744 y(to)549 b(v)-36 b(erify)567 b(\(86\))q(,)577 b(w)-36 b(e)549 b(instead)f(let)g Fr(T)18088 58943 y Fo(")19127 58744 y Fs(b)36 b(e)548 b(the)g(epsilon)h(tub)36 b(e)547 b(o)-36 b(v)g(er)549 b Fq(S)34639 58943 y Fo( )35715 58744 y Fq(\\)373 b Fr(@)72 b(Q)p Fs(.)923 b(Finally)-108 b(,)578 b(the)548 b(estimate)0 60661 y(\()p Fr(\016)1082 60860 y Fp(0)1915 60661 y Fq(\016)307 b Fr(\025)p Fs(\))3999 60440 y Fj(b)4584 60661 y Fs(\()p Fr(t)p Fs(\))397 b(=)h Fr(O)36 b Fs(\()p Fr(t)9879 60171 y Fh(\000)10744 59724 y Fg(\()p Ff(n)p Fe(\000)p Fg(1\))p 10743 59970 2197 40 v 11638 60518 a(2)13128 60661 y Fs(\),)455 b(whic)-36 b(h)450 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1044 x(\000)p Fs(1)222 b Fq(F)14439 71694 y Fo(k)15007 72176 y Fs(\()p Fr(z)59 b Fs(\))17236 69518 y Fj(\023)18435 71391 y Fr(;)20863 69518 y Fj(\022)24161 70594 y Fs(0)27685 69550 y Fq(p)p 28792 69550 V 1044 x(\000)p Fs(1)221 b Fr(e)31299 70793 y Fo(k)22394 71165 y Fq(p)p 23501 71165 V 1043 x(\000)p Fs(1)h Fr(e)26009 71726 y Fo(t)26009 72554 y(k)28111 72208 y Fq(F)29198 71726 y Fo(k)29767 72208 y Fs(\()p Fr(z)59 b Fs(\))32422 69518 y Fj(\023)33842 71391 y Fr(;)1550 b Fs(1)370 b Fq(\024)f Fr(k)414 b Fq(\024)369 b Fr(h)41416 70842 y Fp(2)p Fo(;)p Fp(1)42895 71391 y Fr(;)6432 b Fs(\(93\))p eop %%Page: 36 36 36 35 bop 0 681 a Fm(36)6828 b(MICHAEL)340 b(R.)i(DOUGLAS,)g(BERNARD)e (SHIFFMAN,)i(AND)f(STEVE)f(ZELDITCH)0 3453 y Fl(wher)-66 b(e)478 b Fr(e)4286 3652 y Fo(k)5333 3453 y Fl(is)g(the)h Fr(k)45 b Fl(-th)478 b(standar)-66 b(d)478 b(b)-66 b(asis)478 b(element)f(of)h Fk(C)27448 2971 y Fo(h)27990 2659 y Fg(2)p Ff(;)p Fg(1)29644 3453 y Fl(and)g Fq(F)33268 2971 y Fo(k)33837 3453 y Fs(\()p 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70575 y Fq(j)24617 70093 y Fp(2)p 21014 71169 4129 54 v 21834 72385 a Fs(Im)h Fr(\034)25765 70575 y(@)p 25408 71169 1473 54 v 25408 72385 a(@)72 b(\034)27235 69666 y Fj(Z)27973 72673 y Fo(X)28871 71474 y Fs(\()p Fr(F)475 b Fs(+)363 b(\026)-718 b Fr(\034)148 b(H)103 b Fs(\))295 b Fq(^)g Fs(\012)36816 71673 y Fo(z)37715 71474 y Fs(=)369 b(0)p eop %%Page: 37 37 37 36 bop 11424 681 a Fm(CRITICAL)440 b(POINTS)j(AND)g(SUPERSYMMETRIC)e (V)-114 b(A)-28 b(CUA,)443 b(I)28 b(I)g(I)10843 b(37)0 3337 y Fs(b)-36 b(y)433 b(the)h(critical)g(p)36 b(oin)-36 b(t)433 b(equation)h Fq(r)18196 3536 y Fp(0)18722 3337 y Fr(f)19363 3536 y Fo(G)20152 3337 y Fs(\()p Fr(z)21262 3536 y Fp(0)21788 3337 y Fr(;)221 b(\034)22935 3536 y Fp(0)23461 3337 y Fs(\))369 b(=)f(0.)579 b(This)434 b(pro)-36 b(v)g(es)433 b(\(i\).)1328 4887 y(Next,)i(di\013eren)-36 b(tiating)451 b(\(94\))434 b(with)f Fr(f)19618 5086 y Fo(G)20776 4887 y Fs(=)368 b Fr(f)22797 5086 y Fo(\013)23457 4887 y Fs(,)434 b(w)-36 b(e)434 b(get)11081 7871 y 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b(Gaussian)g(in)-36 b(tegral)458 b(for)0 7987 y Fq(K)1031 7505 y Fp(crit)2963 7987 y Fs(in)463 b(that)g(line)h(resem) -36 b(bles)464 b(to)f(some)h(exten)-36 b(t)464 b(the)f(in)-36 b(tegral)464 b(form)-36 b(ula)464 b(for)g(the)g(exp)36 b(ected)463 b(n)-36 b(um)g(b)36 b(er)462 b(of)0 9536 y(critical)435 b(p)36 b(oin)-36 b(ts)433 b(in)g(spin)g(glass)i(theory) -108 b(,)433 b(whic)-36 b(h)434 b(has)f(exp)36 b(onen)-36 b(tial)434 b(gro)-36 b(wth)434 b(\(see)g(e.g.)579 b([Fy]\).)1328 11086 y(Although)325 b(the)f(naiv)-36 b(e)326 b(coun)-36 b(ting)324 b(of)i(v)-72 b(ariables)326 b(or)f(the)f(analogy)j(to)e (complexit)-36 b(y)326 b(of)g(energy)f(landscap)36 b(es)0 12636 y(bring)633 b(some)h(insigh)-36 b(t)633 b(in)-36 b(to)633 b(v)-72 b(acuum)633 b(coun)-36 b(ting,)683 b(w)-36 b(e)634 b(no)-36 b(w)633 b(p)36 b(oin)-36 b(t)633 b(out)g(some)h (pitfalls)g(in)g(estimating)0 14186 y(n)-36 b(um)g(b)36 b(ers)432 b(of)i(v)-72 b(acua)434 b(or)g(the)f(co)36 b(e\016cien)-36 b(t)434 b Fr(f)142 b Fs(\()p 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b(not)g(a)f(p)36 b(olynomial)515 b(and)d(it)h(is)g(not)f(clear)h(ho)-36 b(w)513 b(to)g(assign)g(it)g(a) 4194 25449 y(`degree')544 b(whic)-36 b(h)544 b(re\015ects)f(its)h(n)-36 b(um)g(b)36 b(er)542 b(of)i(critical)h(p)36 b(oin)-36 b(ts)543 b(on)h(all)g(of)h(T)-108 b(eic)-36 b(hm)36 b(\177)-686 b(uller)543 b(space,)572 b(or)4194 26998 y(equiv)-72 b(alen)-36 b(tly)-108 b(,)533 b(the)512 b(n)-36 b(um)g(b)36 b(er)510 b(of)j(critical)g(p)36 b(oin)-36 b(ts)512 b(in)g Fq(C)590 b Fs(corresp)36 b(onding)511 b(to)i(the)e(\000-orbit)g(of)i Fr(W)181 b Fs(.)4194 28548 y(Examples)434 b(\(e.g.)579 b(in)434 b Fq(x)p Fs(3.6\))g(sho)-36 b(w)434 b(that)f(this)h(n)-36 b(um)g(b)36 b(er)431 b(can)j(b)36 b(e)433 b(relativ)-36 b(ely)436 b(small.)1979 30098 y(\(3\))553 b(It)319 b(seems)f (reasonable)h(to)g(sa)-36 b(y)319 b(that)f(the)f(n)-36 b(um)g(b)36 b(er)317 b(of)i(\015uxes)g(rather)e(than)h(the)g(n)-36 b(um)g(b)36 b(er)317 b(of)i(critical)4194 31648 y(p)36 b(oin)-36 b(ts)488 b(p)36 b(er)488 b(\015ux)g(that)g(dominates)g(the)g (n)-36 b(um)g(b)36 b(er)487 b(of)i(v)-72 b(acua.)743 b(In)488 b(\015ux)g(compacti\014cations,)503 b(the)4194 33197 y(landscap)36 b(e)446 b(should)f(therefore)h(b)36 b(e)445 b(view)-36 b(ed)447 b(as)f(the)f(graph)h(of)g(the)f(scalar)i(p) 36 b(oten)-36 b(tial)446 b Fr(V)47110 33396 y Fo(W)48184 33197 y Fs(\()p Fr(Z)95 b Fs(\))446 b(on)4194 34747 y Fq(C)457 b(\002)379 b(S)100 b Fs(,)587 b(i.e.)948 b(as)556 b(a)h(function)f(of)h(b)36 b(oth)556 b(v)-72 b(ariables)558 b Fr(W)-36 b(;)221 b(Z)95 b Fs(,)588 b(and)556 b(the)g(lo)36 b(cal)558 b(minima)e(should)g(b)36 b(e)4194 36297 y(view)-36 b(ed)434 b(as)g(pairs)g(\()p Fr(W)14975 36496 y Fo(G)15763 36297 y Fr(;)221 b(Z)95 b Fs(\))435 b(with)f Fr(G)369 b Fq(2)f Fr(H)25064 35815 y Fp(3)25589 36297 y Fs(\()p Fr(X)32 b(;)221 b Fk(Z)296 b Fq(\010)30298 35253 y(p)p 31405 35253 1684 54 v 1044 x(\000)p Fs(1)q Fk(Z)p Fs(\))433 b(and)g(with)h Fr(Z)464 b Fq(2)369 b Fr(C)95 b(r)36 b(it)p Fs(\()p Fr(W)47311 36496 y Fo(G)48099 36297 y Fs(\))p Fr(:)1979 37846 y Fs(\(4\))553 b(Ho)-36 b(w)g(ev)g(er)463 b(\(see)g(the)f(problems)g(b)36 b(elo)-36 b(w\))463 b(it)f(is)h(not)f (straigh)-36 b(tforw)g(ard)463 b(to)g(to)f(de\014ne)g(`p)36 b(er)462 b(v)-72 b(acua',)4194 39396 y(since)473 b(the)f(tadp)36 b(ole)473 b(constrain)-36 b(t)472 b(is)i(h)-36 b(yp)36 b(erb)g(olic,)483 b(and)472 b(the)g(total)i(n)-36 b(um)g(b)36 b(er)471 b(of)i(lattice)h(p)36 b(oin)-36 b(ts)472 b(in)4194 40946 y(the)433 b(shell)h(0)369 b Fr(<)g(Q)p Fs([)p Fr(G)p Fs(])h Fr(<)e(L)434 b Fs(is)g(in\014nite.)1979 42496 y(\(5\))553 b(In)565 b(estimating)h Fq(K)13480 42014 y Fp(crit)14948 42496 y Fs(\()p Fr(Z)95 b Fs(\))566 b(w)-36 b(e)566 b(are)f(\014xing)h Fr(Z)661 b Fs(in)566 b(the)f(in)-36 b(terior)565 b(of)h Fq(C)77 b Fs(.)975 b(But)565 b(there)g(could)g (exist)4194 44045 y(singular)446 b(p)36 b(oin)-36 b(ts)446 b(of)h Fq(C)524 b Fs(at)446 b(whic)-36 b(h)446 b Fq(K)22133 43563 y Fp(crit)23601 44045 y Fs(\()p Fr(Z)95 b Fs(\))446 b(blo)-36 b(ws)447 b(up)e(\(see)h([DD1)q(])h(for)f(discussion)g(of)h (conifold)4194 45595 y(p)36 b(oin)-36 b(ts\).)578 b(It)434 b(w)-36 b(ould)433 b(also)i(b)36 b(e)433 b(in)-36 b(teresting)433 b(to)h(study)f Fq(K)31303 45113 y Fp(crit)32771 45595 y Fs(\()p Fr(Z)95 b Fs(\))434 b(as)g Fr(Z)464 b Fq(!)370 b Fr(@)72 b Fq(C)77 b Fs(.)1979 47145 y(\(6\))553 b(As)509 b(men)-36 b(tioned)509 b(in)g Fq(x)p Fs(1.5)h(\(see)f(also)h Fq(x)p Fs(7.3\),)530 b(there)508 b(ma)-36 b(y)510 b(b)36 b(e)509 b(a)g(signi\014can)-36 b(t)509 b(di\013erence)g(b)36 b(et)-36 b(w)g(een)4194 48695 y(the)430 b(order)g(of)i(magnitude)e(of)h (the)f(densit)-36 b(y)431 b(of)g(critical)h(p)36 b(oin)-36 b(ts)430 b(and)g(of)i(the)e(n)-36 b(um)g(b)36 b(er)429 b(of)i(critical)4194 50244 y(p)36 b(oin)-36 b(ts,)418 b(since)c Fq(C)492 b Fs(is)415 b(an)f(incomplete)g(K\177)-650 b(ahler)415 b(manifold)g(of)f(p)36 b(ossibly)415 b(quite)g(small)g(v) -36 b(olume.)572 b(See)4194 51794 y([LS])394 b(for)g(the)g(curren)-36 b(t)392 b(state)i(of)h(the)e(art)h(on)g(the)g(v)-36 b(olume.)565 b(There)394 b(is)g(no)g(analogue)h(of)g(the)e(small)4194 53344 y(v)-36 b(olume)434 b(of)h(the)e(con\014guration)g(space)g(in)h (spin)f(glass)i(complexit)-36 b(y)-108 b(.)1979 54894 y(\(7\))553 b(The)517 b(tadp)36 b(ole)517 b(constrain)-36 b(t)517 b(\(1\))f(b)36 b(ecomes)518 b(m)-36 b(uc)g(h)515 b(more)i(highly)h(constraining)f(as)g(the)g(n)-36 b(um)g(b)36 b(er)4194 56443 y Fr(b)4747 56642 y Fp(3)5701 56443 y Fs(of)430 b(v)-72 b(ariables)429 b(increases)g(for)g(\014xed)f Fr(L)g Fs(and)g(is)h(resp)36 b(onsible)428 b(for)h(the)f(factor)43705 55920 y Fp(1)p 43011 56138 1857 54 v 43011 56901 a(\()p Fo(b)43780 57024 y Fg(3)44241 56901 y Fp(\)!)45430 56443 y Fs(in)g(Theorem)4194 58156 y(1.4.)579 b(Again,)435 b(no)e(suc)-36 b(h)433 b(feature)h(exists)g(in)f(complexit)-36 b(y)435 b(estimates)f(in)g(spin)f(glasses.)0 61054 y(7.2.)1396 b FA(Problems.)652 b Fs(The)457 b(issues)g(men)-36 b(tioned)456 b(ab)36 b(o)-36 b(v)g(e)458 b(\(and)e(the)g(detailed)h(heuristics)g(in) g Fq(x)p Fs(7.3\))h(suggest)0 62604 y(a)434 b(n)-36 b(um)g(b)36 b(er)432 b(of)i(problems.)578 b(The)434 b(ultimate)f(goal)i(is:)0 65093 y Ft(Pr)-30 b(oblem)661 b FA(7.1.)625 b Fl(Do)-66 b(es)594 b(string)f(the)-66 b(ory)594 b(c)-66 b(ontain)592 b(a)j(vacuum)f(c)-66 b(onsistent)592 b(with)i(the)g(standar)-66 b(d)594 b(mo)-66 b(del,)0 66642 y(and)595 b(if)f(so,)628 b(how)595 b(many?)988 b(Find)595 b(examples)f(of)h(Calabi-Y)-100 b(au)595 b(manifolds,)626 b(and)595 b(any)g(other)g(p)-66 b(ostulate)g(d)0 68192 y(structur)g(es,)464 b(for)h(which)g(it)f(is)h (c)-66 b(ertain)463 b(that)i(such)g(a)g(vacuum)g(exists.)1328 70681 y Fs(No)-36 b(w)452 b(testing)f(consistency)g(with)g(the)g (standard)f(mo)36 b(del)451 b(requires)g(elucidating)h(far)f(more)g (structure)0 72231 y(of)494 b(a)g(candidate)f(v)-72 b(acuum)494 b({)g(the)f(gauge)h(group,)508 b(the)493 b(matter)h(con)-36 b(ten)g(t,)508 b(and)493 b(so)h(forth)f({)h(than)f(w)-36 b(e)494 b(are)p eop 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42 42 41 bop 0 681 a Fm(42)6828 b(MICHAEL)340 b(R.)i(DOUGLAS,)g(BERNARD)e (SHIFFMAN,)i(AND)f(STEVE)f(ZELDITCH)1979 3337 y Fs(\(1\))553 b(the)539 b(n)-36 b(um)g(b)36 b(er)538 b(of)i(\015uxes)f Fr(G)g Fs(satisfying)i(the)e(tadp)36 b(ole)539 b(constrain)-36 b(t)539 b(with)h(a)f(critical)i(p)36 b(oin)-36 b(t)538 b(in)i(a)4194 4887 y(compact)434 b(subset)e Fq(K)389 b(\032)369 b(C)77 b Fs(;)1979 6437 y(\(2\))553 b(the)451 b(n)-36 b(um)g(b)36 b(er)449 b(of)j(critical)g(p)36 b(oin)-36 b(ts)451 b(`p)36 b(er)451 b(\015ux',)456 b(or)451 b(more)g(precisely)h (p)36 b(er)451 b(\000-orbit)f(of)i(\015uxes,)k(in)451 b Fq(K)4194 7987 y Fs(\(see)434 b Fq(x)p Fs(3.6)h(to)e(clarify)j(this)d (distinction\);)1979 9536 y(\(3\))553 b(the)386 b(total)h(n)-36 b(um)g(b)36 b(er)384 b(of)j(critical)g(p)36 b(oin)-36 b(ts)386 b(in)g Fq(K)405 b Fs(of)387 b(all)g(\015uxes)f(satisfying)i (the)e(tadp)36 b(ole)386 b(constrain)-36 b(t.)1328 11468 y(W)-108 b(e)434 b(can)f(de\014ne)g(the)g(\014rst)g(quan)-36 b(tit)g(y)433 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b(metric)g(relativ)-36 b(e)423 b(to)f(the)f(W)-108 b(eil-P)-36 b(etersson)422 b(metric)p eop %%Page: 44 44 44 43 bop 0 681 a Fm(44)6828 b(MICHAEL)340 b(R.)i(DOUGLAS,)g(BERNARD)e (SHIFFMAN,)i(AND)f(STEVE)f(ZELDITCH)0 3337 y Fs(satisfy)538 b Fr(\026)4947 3536 y Fp(max)6753 3337 y Fs(\()p Fr(Z)95 b Fs(\))544 b Fq(\024)h Fr(\026)p Fs(.)886 b(W)-108 b(e)537 b(ha)-36 b(v)g(e)536 b(b)36 b(ounded)535 b(the)h(determinan)-36 b(t)536 b(of)h(\003)f(b)-36 b(y)536 b(a)h(p)36 b(o)-36 b(w)g(er)537 b(of)g(an)f(extremal)0 4887 y(eigen)-36 b(v)-72 b(alue,)430 b(but)d(it)i(could)f(also)h(b)36 b(e)428 b(iden)-36 b(ti\014ed)427 b(with)h(the)g(v)-36 b(olume)428 b(densit)-36 b(y)428 b(of)h(the)f(Ho)36 b(dge)428 b(metric.)577 b(W)-108 b(e)0 6437 y(note)533 b(that)f(the)g(lo)-36 b(w)g(er)533 b(b)36 b(ound)532 b(tends)g(to)h(zero)g(and)f(the)g(upp)36 b(er)532 b(b)36 b(ound)531 b(tends)h(to)h(in\014nit)-36 b(y)532 b(in)h Fq(\030)538 b(\006)p Fr(b)51474 6636 y Fp(3)0 7987 y Fs(p)36 b(o)-36 b(w)g(ers)462 b(of)h 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