This is TeX, Version 3.14159 (Web2C 7.3.1) (format=latex 2003.2.18) 29 SEP 2003 14:27 **anal_4 (anal_4.tex LaTeX2e <2001/06/01> Babel and hyphenation patterns for american, french, german, ngerman, i talian, nohyphenation, loaded. (/usr/local/share/texmf/tex/latex/base/article.cls Document Class: article 2001/04/21 v1.4e Standard LaTeX document class (/usr/local/share/texmf/tex/latex/base/size10.clo File: size10.clo 2001/04/21 v1.4e Standard LaTeX file (size option) ) \c@part=\count79 \c@section=\count80 \c@subsection=\count81 \c@subsubsection=\count82 \c@paragraph=\count83 \c@subparagraph=\count84 \c@figure=\count85 \c@table=\count86 \abovecaptionskip=\skip41 \belowcaptionskip=\skip42 \bibindent=\dimen102 ) (/usr/local/share/texmf/tex/latex/amsmath/amsmath.sty Package: amsmath 2000/07/18 v2.13 AMS math features \@mathmargin=\skip43 For additional information on amsmath, use the `?' option. (/usr/local/share/texmf/tex/latex/amsmath/amstext.sty Package: amstext 2000/06/29 v2.01 (/usr/local/share/texmf/tex/latex/amsmath/amsgen.sty File: amsgen.sty 1999/11/30 v2.0 \@emptytoks=\toks14 \ex@=\dimen103 )) (/usr/local/share/texmf/tex/latex/amsmath/amsbsy.sty Package: amsbsy 1999/11/29 v1.2d \pmbraise@=\dimen104 ) (/usr/local/share/texmf/tex/latex/amsmath/amsopn.sty Package: amsopn 1999/12/14 v2.01 operator names ) \inf@bad=\count87 LaTeX Info: Redefining \frac on input line 211. \uproot@=\count88 \leftroot@=\count89 LaTeX Info: Redefining \overline on input line 307. \classnum@=\count90 \DOTSCASE@=\count91 LaTeX Info: Redefining \ldots on input line 379. LaTeX Info: Redefining \dots on input line 382. LaTeX Info: Redefining \cdots on input line 467. \Mathstrutbox@=\box26 \strutbox@=\box27 \big@size=\dimen105 LaTeX Font Info: Redeclaring font encoding OML on input line 567. LaTeX Font Info: Redeclaring font encoding OMS on input line 568. \macc@depth=\count92 \c@MaxMatrixCols=\count93 \dotsspace@=\muskip10 \c@parentequation=\count94 \dspbrk@lvl=\count95 \tag@help=\toks15 \row@=\count96 \column@=\count97 \maxfields@=\count98 \andhelp@=\toks16 \eqnshift@=\dimen106 \alignsep@=\dimen107 \tagshift@=\dimen108 \tagwidth@=\dimen109 \totwidth@=\dimen110 \lineht@=\dimen111 \@envbody=\toks17 \multlinegap=\skip44 \multlinetaggap=\skip45 \mathdisplay@stack=\toks18 LaTeX Info: Redefining \[ on input line 2666. LaTeX Info: Redefining \] on input line 2667. ) (/usr/local/share/texmf/tex/latex/amsfonts/amssymb.sty Package: amssymb 2002/01/22 v2.2d (/usr/local/share/texmf/tex/latex/amsfonts/amsfonts.sty Package: amsfonts 2001/10/25 v2.2f \symAMSa=\mathgroup4 \symAMSb=\mathgroup5 LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold' (Font) U/euf/m/n --> U/euf/b/n on input line 132. )) (/usr/local/share/texmf/tex/latex/jknappen/mathrsfs.sty Package: mathrsfs 1996/01/01 Math RSFS package v1.0 (jk) \symrsfs=\mathgroup6 ) (anal_4.aux) \openout1 = `anal_4.aux'. LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 10. LaTeX Font Info: ... okay on input line 10. LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 10. LaTeX Font Info: ... okay on input line 10. LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 10. LaTeX Font Info: ... okay on input line 10. LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 10. 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(/usr/local/share/texmf/tex/latex/jknappen/ursfs.fd File: ursfs.fd 1998/03/24 rsfs font definition file (jk) ) Overfull \hbox (3.64536pt too wide) in paragraph at lines 19--25 \OT1/cmr/m/n/10 We show that for all $\OML/cmm/m/it/10 t \OMS/cmsy/m/n/10 2 \OT 1/cmr/m/n/10 [\OML/cmm/m/it/10 T[]; R\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 ;$ \OT1/ cmr/m/n/10 we have $\OMS/cmsy/m/n/10 j\OML/cmm/m/it/10 y[]\OT1/cmr/m/n/10 (\OML /cmm/m/it/10 t\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ^^@ \OML/cmm/m/it/10 y[]\OT1/c mr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )\OMS/cmsy/m/n/10 j ^^T j\OML/cmm/ m/it/10 y[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 T[]\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n /10 ^^@ \OML/cmm/m/it/10 y[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 T[]\OT1/cmr/m/n/1 0 )\OMS/cmsy/m/n/10 j\OML/cmm/m/it/10 e[]:$ [] [1 ] Overfull \hbox (0.84637pt too wide) in paragraph at lines 45--63 \OT1/cmr/m/n/10 Then by the lo-cal Lip-s-chitz prop-erty there are $\OML/cmm/m/ it/10 ^^O; K$ \OT1/cmr/m/n/10 such that for $(\OML/cmm/m/it/10 t[]; y[]\OT1/cmr /m/n/10 )\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t[]; y[]\OT1/cmr/ m/n/10 ) \OMS/cmsy/m/n/10 2 [] [2] [3] [4] Overfull \hbox (10.77649pt too wide) detected at line 142 \OML/cmm/m/it/10 z[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 ) = \OML /cmm/m/it/10 e[]\OT1/cmr/m/n/10 (\OMS/cmsy/m/n/10 ^^@\OT1/cmr/m/n/10 (\OML/cmm/ m/it/10 M \OT1/cmr/m/n/10 + \OML/cmm/m/it/10 L\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 z\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 ) + \OML/cmm/m/it/10 z[]\O T1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )) \OMS/cmsy/m/n/10 ^^U ^^@\OT 1/cmr/m/n/10 (\OML/cmm/m/it/10 M \OT1/cmr/m/n/10 + \OML/cmm/m/it/10 L\OT1/cmr/m /n/10 )\OML/cmm/m/it/10 e[]z\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 ) = \OMS/cmsy/m/n/10 ^^@\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 M \OT1/cmr/m/n/10 + \ OML/cmm/m/it/10 L\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 z[]\OT1/cmr/m/n/10 (\OML/cmm /m/it/10 t\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 : [] Overfull \hbox (14.26595pt too wide) detected at line 148 \OML/cmm/m/it/10 z\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 ) = \OML/c mm/m/it/10 z\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )[] = [] \OMS/cm sy/m/n/10 ^^T [] \OT1/cmr/m/n/10 = \OML/cmm/m/it/10 z\OT1/cmr/m/n/10 (\OML/cmm/ m/it/10 c\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 e[] \OMS/cmsy/m/n/10 ^^T \OT1/cmr/m/ n/10 (\OMS/cmsy/m/n/10 j\OML/cmm/m/it/10 Y[]\OMS/cmsy/m/n/10 j[] \OT1/cmr/m/n/1 0 + 1)\OML/cmm/m/it/10 e[]: [] [5] [6] [7] [8] [9] [10] (anal_4.aux) ) Here is how much of TeX's memory you used: 1146 strings out of 20880 12136 string characters out of 196184 63158 words of memory out of 350001 4110 multiletter control sequences out of 10000+15000 14038 words of font info for 56 fonts, out of 400000 for 1000 14 hyphenation exceptions out of 10000 27i,8n,24p,539b,248s stack positions out of 3000i,100n,1500p,50000b,4000s Output written on anal_4.dvi (10 pages, 41200 bytes).