This is TeX, Version 3.14159 (Web2C 7.3) (format=latex 1999.4.27) 23 MAY 2000 10:14 **clay.tex (clay.tex LaTeX2e <1998/12/01> Babel and hyphenation patterns for american, french, german, ngerman, n ohyphenation, loaded. (/net/fern/m4/tex/teTeX-0.9/share/texmf/tex/latex/base/latex209.def File: latex209.def 1998/05/13 v0.52 Standard LaTeX file Entering LaTeX 2.09 COMPATIBILITY MODE ************************************************************* !!WARNING!! !!WARNING!! !!WARNING!! !!WARNING!! This mode attempts to provide an emulation of the LaTeX 2.09 author environment so that OLD documents can be successfully processed. It should NOT be used for NEW documents! New documents should use Standard LaTeX conventions and start with the \documentclass command. Compatibility mode is UNLIKELY TO WORK with LaTeX 2.09 style files that change any internal macros, especially not with those that change the FONT SELECTION or OUTPUT ROUTINES. 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LaTeX Info: Redefining \cal on input line 488. LaTeX Info: Redefining \mit on input line 489. \bibindent=\dimen105 ) (/net/fern/m4/tex/teTeX-0.9/share/texmf/tex/latex/amsfonts/amssymb.sty Package: amssymb 1996/11/03 v2.2b (/net/fern/m4/tex/teTeX-0.9/share/texmf/tex/latex/amsfonts/amsfonts.sty Package: amsfonts 1997/09/17 v2.2e \@emptytoks=\toks14 \symAMSa=\mathgroup11 \symAMSb=\mathgroup12 LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold' (Font) U/euf/m/n --> U/euf/b/n on input line 133. )) (/net/fern/m4/tex/teTeX-0.9/share/texmf/tex/generic/misc/psfig.sty \@unused=\write3 \ps@stream=\read1 \p@intvaluex=\dimen106 \p@intvaluey=\dimen107 psfig/tex 1.10-dvips ) (/u/sonia/h1/fa/tao/tex/MathMacro.tex) \c@theorem=\count91 \c@proposition=\count92 \c@lemma=\count93 \c@corollary=\count94 \c@remark=\count95 (clay.aux) \openout1 = `clay.aux'. LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 42. LaTeX Font Info: ... okay on input line 42. 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(/net/fern/m4/tex/teTeX-0.9/share/texmf/tex/latex/amsfonts/umsb.fd File: umsb.fd 1995/01/05 v2.2e AMS font definitions ) [1 ] LaTeX Font Info: Try loading font information for OMS+cmr on input line 71. (/net/fern/m4/tex/teTeX-0.9/share/texmf/tex/latex/base/omscmr.fd File: omscmr.fd 1998/03/27 v2.5g Standard LaTeX font definitions ) LaTeX Font Info: Font shape `OMS/cmr/m/n' in size <17.28> not available (Font) Font shape `OMS/cmsy/m/n' tried instead on input line 71. Overfull \hbox (0.79863pt too wide) in paragraph at lines 73--74 \OT1/cmr/m/n/17.28 area, while a more so-phis-ti-cated ``three-point U-turn'' [] Overfull \hbox (0.3179pt too wide) in paragraph at lines 77--78 []\OT1/cmr/m/n/17.28 Firstly, one can trans-late a nee-dle for ar-bi-trar-ily s mall [] Overfull \hbox (0.40561pt too wide) in paragraph at lines 79--80 []\OT1/cmr/m/n/17.28 Secondly, one can find a col-lec-tion of rect-an-gles whos e [] psfig: searching besicovitch.eps for bounding box psfig: including besicovitch.eps [2] Overfull \hbox (14.3623pt too wide) in paragraph at lines 85--86 \OT1/cmr/m/n/17.28 dorff and Minkowski di-men-sion ex-actly 2 (Drury, 1971). [] [3] Overfull \hbox (0.4954pt too wide) in paragraph at lines 89--90 \OT1/cmr/m/n/17.28 and Minkowski di-men-sion $\OML/cmm/m/it/17.28 n$\OT1/cmr/m/ n/17.28 . On the dis-cretized level, [] Overfull \hbox (9.07585pt too wide) in paragraph at lines 91--92 \OT1/cmr/m/n/17.28 higher di-men-sions it is still open. Much re-cent progress [] Overfull \hbox (8.94313pt too wide) in paragraph at lines 92--94 \OT1/cmr/m/n/17.28 (Katz-Laba-T 1999, Katz-T 1999, Laba-T 2000), while [] [4] [5] Overfull \hbox (23.59973pt too wide) in paragraph at lines 117--118 \OT1/cmr/m/n/17.28 lead to some in-ter-est-ing near-counterexamples to Kakeya [] Overfull \hbox (8.99448pt too wide) in paragraph at lines 117--118 \OT1/cmr/m/n/17.28 such as the Fal-coner dis-tance prob-lem and the Fursten- [] [6] [7] [8] psfig: searching fefferman.eps for bounding box psfig: including fefferman.eps [9] [10] Overfull \hbox (37.71582pt too wide) in paragraph at lines 164--166 \OT1/cmr/m/n/17.28 The \OT1/cmr/m/it/17.28 Bochner-Riesz con-jec-ture \OT1/cmr/ m/n/17.28 as-serts that these weighted [] Overfull \hbox (11.81616pt too wide) in paragraph at lines 167--168 \OT1/cmr/m/n/17.28 Riesz con-jec-ture im-plies the Kakeya con-jec-ture. (Bour- [] [11] Overfull \hbox (24.34421pt too wide) in paragraph at lines 173--174 \OT1/cmr/m/n/17.28 would im-ply the Kakeya con-jec-ture just like the Bochner- [] [12] [13] [14] [15] psfig: searching smoothing-counter.eps for bounding box psfig: including smoothing-counter.eps [16] Overfull \hbox (4.93185pt too wide) in paragraph at lines 223--224 []\OT1/cmr/m/n/17.28 Variations of this ex-am-ple show that the lo-cal smooth- [] [17] [18] Overfull \hbox (25.84143pt too wide) detected at line 242 \OT1/cmr/m/n/17.28 (\OML/cmm/m/it/17.28 u[]\OT1/cmr/m/n/17.28 )[] \OMS/cmsy/m/n /17.28 ^^@ \OT1/cmr/m/n/17.28 ^^A\OML/cmm/m/it/17.28 u[] \OT1/cmr/m/n/17.28 = 0 ; \OML/cmm/m/it/17.28 u[]\OT1/cmr/m/n/17.28 (0\OML/cmm/m/it/17.28 ; x\OT1/cmr/ m/n/17.28 ) = \OML/cmm/m/it/17.28 f\OT1/cmr/m/n/17.28 (\OML/cmm/m/it/17.28 x\OT 1/cmr/m/n/17.28 ); (\OML/cmm/m/it/17.28 u[]\OT1/cmr/m/n/17.28 )[](0\OML/cmm/m/ it/17.28 ; x\OT1/cmr/m/n/17.28 ) = \OML/cmm/m/it/17.28 g\OT1/cmr/m/n/17.28 (\OM L/cmm/m/it/17.28 x\OT1/cmr/m/n/17.28 ) [] Overfull \hbox (10.47067pt too wide) detected at line 244 \OT1/cmr/m/n/17.28 (\OML/cmm/m/it/17.28 u[]\OT1/cmr/m/n/17.28 )[] \OMS/cmsy/m/n /17.28 ^^@ \OT1/cmr/m/n/17.28 ^^A\OML/cmm/m/it/17.28 u[] \OT1/cmr/m/n/17.28 = ( \OML/cmm/m/it/17.28 u[]\OT1/cmr/m/n/17.28 )[]; \OML/cmm/m/it/17.28 u[]\OT1/cmr /m/n/17.28 (0\OML/cmm/m/it/17.28 ; x\OT1/cmr/m/n/17.28 ) = 0; (\OML/cmm/m/it/1 7.28 u[]\OT1/cmr/m/n/17.28 )[](0\OML/cmm/m/it/17.28 ; x\OT1/cmr/m/n/17.28 ) = 0 \OML/cmm/m/it/17.28 : [] [19] [20] Overfull \hbox (38.20456pt too wide) detected at line 261 \OMS/cmsy/m/n/17.28 k\OML/cmm/m/it/17.28 u[]v[] \OMS/cmsy/m/n/17.28 ^^@ r\OML/c mm/m/it/17.28 u \OMS/cmsy/m/n/17.28 ^^A r\OML/cmm/m/it/17.28 v\OMS/cmsy/m/n/17. 28 k[] ^^T \OML/cmm/m/it/17.28 C\OMS/cmsy/m/n/17.28 kr[]\OML/cmm/m/it/17.28 f[] \OMS/cmsy/m/n/17.28 k[]kr\OML/cmm/m/it/17.28 f[]\OMS/cmsy/m/n/17.28 k[] [] Overfull \hbox (5.65416pt too wide) in paragraph at lines 267--268 \OT1/cmr/m/n/17.28 deriva-tives (much work by Klainerman-Machedon and [] [21] Overfull \hbox (34.04495pt too wide) in paragraph at lines 269--270 []\OT1/cmr/m/n/17.28 Bilinear $\OML/cmm/m/it/17.28 L[]$ \OT1/cmr/m/n/17.28 es-t i-mates are usu-ally proven us-ing Plancherel's [] Overfull \hbox (1.7945pt too wide) in paragraph at lines 271--272 []\OT1/cmr/m/n/17.28 Very re-cently (Tataru, Bahouri-Chemin, Klain-er-man, [] [22] [23] (clay.aux) ) Here is how much of TeX's memory you used: 1192 strings out of 10896 11800 string characters out of 71883 54934 words of memory out of 263001 4122 multiletter control sequences out of 10000+0 22736 words of font info for 87 fonts, out of 200000 for 1000 15 hyphenation exceptions out of 1000 25i,5n,24p,735b,248s stack positions out of 300i,100n,500p,30000b,4000s Output written on clay.dvi (23 pages, 31552 bytes).