PTITeX 3.4, (c) Personal TeX, Inc. 1985-1998 This is TeX 3.14159 (preloaded format=LATEX 0.11.30) 11 JAN 2001 12:21 **&LATEX C:/FPhandbook/hyperconvex.tex (C:\FPhandbook\hyperconvex.tex LaTeX2e <1998/12/01> Babel and hyphenation patterns for american, loaded. (C:\PCTeXv4\AMSLTX12\amsart.cls Document Class: amsart 1995/02/23 v1.2b \linespacing=\dimen102 \normalparindent=\dimen103 \normaltopskip=\skip41 (C:\PCTeXv4\AMSLTX12\amsgen.sty File: amsgen.sty 1995/02/01 v1.2a \@emptytoks=\toks14 \ex@=\dimen104 ) (C:\PCTeXv4\AMSLTX12\amsmath.sty Package: amsmath 1995/02/23 v1.2b AMS math features \@mathmargin=\skip42 (C:\PCTeXv4\AMSLTX12\amstext.sty Package: amstext 1995/01/25 v1.2 ) (C:\PCTeXv4\AMSLTX12\amsbsy.sty Package: amsbsy 1995/01/20 v1.2 \pmbraise@=\dimen105 ) (C:\PCTeXv4\AMSLTX12\amsopn.sty Package: amsopn 1995/02/20 v1.2a operator names ) LaTeX Info: Redefining \frac on input line 180. \uproot@=\count79 \leftroot@=\count80 \classnum@=\count81 \DOTSCASE@=\count82 LaTeX Info: Redefining \dots on input line 327. LaTeX Info: Redefining \ldots on input line 410. LaTeX Info: Redefining \cdots on input line 414. \Mathstrutbox@=\box26 \strutbox@=\box27 \big@size=\dimen106 LaTeX Font Info: Redeclaring font encoding OML on input line 492. LaTeX Font Info: Redeclaring font encoding OMS on input line 493. \skewcharcount@=\count83 \familycount@=\count84 \pointcount@=\count85 \accentdimen@=\dimen107 \accentmu@=\count86 \minaw@=\dimen108 \c@MaxMatrixCols=\count87 \dotsspace@=\muskip10 \c@parentequation=\count88 \dspbrk@lvl=\count89 \tag@help=\toks15 \row@=\count90 \column@=\count91 \maxfields@=\count92 \andhelp@=\toks16 \eqnshift@=\dimen109 \alignsep@=\dimen110 \tagshift@=\dimen111 \tagwidth@=\dimen112 \totwidth@=\dimen113 \lineht@=\dimen114 \@envbody=\toks17 \multlinegap=\skip43 \multlinetaggap=\skip44 ) (C:\PCTeXv4\AMSLTX12\amsfonts.sty Package: amsfonts 1995/02/01 v2.2b \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 128. ) \copyins=\insert233 \abstractbox=\box28 \listisep=\skip45 \c@part=\count93 \c@section=\count94 \c@subsection=\count95 \c@subsubsection=\count96 \c@paragraph=\count97 \c@figure=\count98 \c@table=\count99 \abovecaptionskip=\skip46 \belowcaptionskip=\skip47 (C:\PCTeXv4\AMSLTX12\amsthm.sty Package: amsthm 1995/02/01 v1.2a \theorem@style=\toks18 \theorembodyfont=\toks19 \theoremheadfont=\toks20 \theoremnotefont=\toks21 \theoremheadpunct=\toks22 \theorempreskipamount=\skip48 \theorempostskipamount=\skip49 )) (C:\PCTeXv4\AMSLTX12\amssymb.sty Package: amssymb 1995/02/01 v2.2a ) (C:\PCTeXv4\LATEX2E\newlfont.sty Package: newlfont 1998/08/17 v2.2m Standard LaTeX package LaTeX Info: Redefining \em on input line 49. (C:\PCTeXv4\LATEX2E\latexsym.sty Package: latexsym 1998/08/17 v2.2e Standard LaTeX package (lasy symbols) \symlasy=\mathgroup6 LaTeX Font Info: Overwriting symbol font `lasy' in version `bold' (Font) U/lasy/m/n --> U/lasy/b/n on input line 29. )) (C:\PCTeXv4\LATEX2E\inputenc.sty beta test version Package: inputenc 1998/03/05 v0.97 Input encoding file (test version: still lia ble to change) (C:\PCTeXv4\LATEX2E\ansinew.def File: ansinew.def 1998/03/05 v0.97 Input encoding file (test version: still lia ble to change) )) (C:\PCTeXv4\LATEX2E\graphicx.sty Package: graphicx 1997/06/09 v1.0d Enhanced LaTeX Graphics (DPC,SPQR) (C:\PCTeXv4\LATEX2E\keyval.sty Package: keyval 1998/06/05 v1.11 key=value parser (DPC) \KV@toks@=\toks23 ) (C:\PCTeXv4\LATEX2E\graphics.sty Package: graphics 1998/05/27 v1.0h Standard LaTeX Graphics (DPC,SPQR) (C:\PCTeXv4\LATEX2E\trig.sty Package: trig 1994/10/16 v1.08 sin cos tan (DPC) ) (C:\PCTeXv4\LATEX2E\graphics.cfg) Package graphics Info: Driver file: dvips.def on input line 85. (C:\PCTeXv4\LATEX2E\dvips.def File: dvips.def 1998/05/03 v3.0f Driver-dependant file (DPC,SPQR) )) \Gin@req@height=\dimen115 \Gin@req@width=\dimen116 ) \c@theorem=\count100 ! LaTeX Error: Command \proof already defined. Or name \end... illegal, see p.192 of the manual. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.38 ...xtbf{#1.} }{\ \rule{0.5em}{0.5em}} ? \@indexfile=\write3 Writing index file hyperconvex.idx LaTeX Warning: Unused global option(s): [showkey]. (C:/FPhandbook/hyperconvex.aux) LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 45. LaTeX Font Info: ... okay on input line 45. LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 45. LaTeX Font Info: ... okay on input line 45. LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 45. LaTeX Font Info: ... okay on input line 45. LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 45. LaTeX Font Info: ... okay on input line 45. 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(C:\PCTeXv4\LATEX2E\ulasy.fd File: ulasy.fd 1998/08/17 v2.2eLaTeX symbol font definitions ) (C:/FPhandbook/hyperconvex.toc) \tf@toc=\write4 Underfull \hbox (badness 10000) in paragraph at lines 78--91 [] Underfull \hbox (badness 10000) in paragraph at lines 92--102 [] Underfull \hbox (badness 10000) in paragraph at lines 103--113 [] Underfull \hbox (badness 10000) in paragraph at lines 114--144 [] [1] Underfull \hbox (badness 10000) in paragraph at lines 145--172 [] Underfull \hbox (badness 10000) in paragraph at lines 191--200 [] Underfull \vbox (badness 10000) has occurred while \output is active [] Overfull \vbox (1.63055pt too high) has occurred while \output is active [] [2] Underfull \hbox (badness 10000) in paragraph at lines 259--263 [] Underfull \hbox (badness 10000) in paragraph at lines 266--268 [] Underfull \hbox (badness 10000) in paragraph at lines 271--276 [] Underfull \hbox (badness 10000) in paragraph at lines 280--286 [] Underfull \hbox (badness 10000) in paragraph at 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(badness 10000) in paragraph at lines 2330--2334 [] Underfull \hbox (badness 10000) in paragraph at lines 2345--2349 [] Underfull \hbox (badness 10000) in paragraph at lines 2366--2371 [] [27] Underfull \hbox (badness 10000) in paragraph at lines 2379--2382 [] Overfull \vbox (1.63055pt too high) has occurred while \output is active [] [28] [29] Underfull \hbox (badness 10000) in paragraph at lines 2523--2527 [] Underfull \hbox (badness 10000) in paragraph at lines 2529--2539 [] Underfull \hbox (badness 10000) in paragraph at lines 2554--2557 [] Package amsfonts Warning: Obsolete command \frak; \mathfrak should be used inst ead on input line 2576. LaTeX Font Info: Try loading font information for U+euf on input line 2576. (C:\PCTeXv4\AMSLTX12\ueuf.fd File: Ueuf.fd 1995/01/05 v2.2e AMS font definitions ) Overfull \vbox (1.63055pt too high) has occurred while \output is active [] [30] Underfull \hbox (badness 10000) in paragraph at lines 2640--2649 [] Underfull \hbox (badness 10000) in paragraph at lines 2654--2658 [] Overfull \hbox (1.1535pt too wide) in paragraph at lines 2683--2689 []\OT1/cmr/m/it/10 Let $\OML/cmm/m/it/10 H$ \OT1/cmr/m/it/10 be hy-per-con-vex, let $\OML/cmm/m/it/10 T[] \OT1/cmr/m/n/10 : \OML/cmm/m/it/10 H \OMS/cmsy/m/n/1 0 ! E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 H\OT1/cmr/m/n/10 )$ \OT1/cmr/m/it/10 be non-ex-pan-sive and sup-pose $[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 T[]\OT1/cmr/m /n/10 ) \OMS/cmsy/m/n/10 6\OT1/cmr/m/n/10 = [] [31] Underfull \hbox (badness 10000) in paragraph at lines 2755--2763 [] Underfull \hbox (badness 10000) in paragraph at lines 2764--2784 [] Underfull \hbox (badness 10000) in paragraph at lines 2791--2796 [] Overfull \vbox (1.63055pt too high) has occurred while \output is active [] [32] Underfull \hbox (badness 10000) in paragraph at lines 2842--2848 [] Underfull \hbox (badness 10000) in paragraph at lines 2865--2873 [] Underfull \hbox (badness 10000) in paragraph at lines 2885--2888 [] Underfull \hbox (badness 10000) in paragraph at lines 2895--2900 [] [33] Underfull \hbox (badness 10000) in paragraph at lines 2937--2941 [] Underfull \hbox (badness 10000) in paragraph at lines 2947--2951 [] Underfull \hbox (badness 10000) in paragraph at lines 2959--2968 [] Underfull \hbox (badness 10000) in paragraph at lines 2971--2978 [] Overfull \vbox (1.63055pt too high) has occurred while \output is active [] [34] Underfull \hbox (badness 10000) in paragraph at lines 3015--3020 [] Underfull \hbox (badness 10000) in paragraph at lines 3030--3034 [] Underfull \hbox (badness 10000) in paragraph at lines 3039--3043 [] Underfull \hbox (badness 10000) in paragraph at lines 3060--3064 [] Underfull \hbox (badness 10000) in paragraph at lines 3076--3080 [] [35] Underfull \hbox (badness 10000) in paragraph at lines 3111--3118 [] Underfull \hbox (badness 10000) in paragraph at lines 3125--3126 [] Overfull \hbox (2.77399pt too wide) in paragraph at lines 3163--3169 \OT1/cmr/m/n/10 Therefore, $\OML/cmm/m/it/10 r\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 c[]\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 62 [] \OML/cmm/m/it/10 G\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x[]\OT1/cmr/m/n/10 )$ and $\OML/cmm/m/it/10 r\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 c[]\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 2 []\OT1/cmr/m/n/10 (\O MS/cmsy/m/n/10 f\OML/cmm/m/it/10 x[]\OT1/cmr/m/n/10 ; \OML/cmm/m/it/10 i \OMS/c msy/m/n/10 2 \OML/cmm/m/it/10 I\OMS/cmsy/m/n/10 g\OT1/cmr/m/n/10 )$, con-tra-di ct-ing the as-sump-tion $[](\OMS/cmsy/m/n/10 f\OML/cmm/m/it/10 x[]\OT1/cmr/m/n/ 10 ; \OML/cmm/m/it/10 i \OMS/cmsy/m/n/10 2 [] Underfull \hbox (badness 10000) in paragraph at lines 3163--3169 [] Overfull \vbox (1.63055pt too high) has occurred while \output is active [] [36] Underfull \hbox (badness 10000) in paragraph at lines 3183--3184 [] Underfull \hbox (badness 10000) in paragraph at lines 3188--3192 [] Underfull \hbox (badness 10000) in paragraph at lines 3207--3208 [] Underfull \hbox (badness 10000) in paragraph at lines 3214--3222 [] Underfull \hbox (badness 10000) in paragraph at lines 3233--3240 [] Underfull \hbox (badness 10000) in paragraph at lines 3242--3249 [] [37] Underfull \hbox (badness 10000) in paragraph at lines 3277--3281 [] Underfull \hbox (badness 10000) in paragraph at lines 3290--3300 [] Underfull \hbox (badness 10000) in paragraph at lines 3311--3315 [] Underfull \hbox (badness 10000) in paragraph at lines 3319--3323 [] Underfull \hbox (badness 10000) in paragraph at lines 3337--3341 [] Overfull \vbox (1.63055pt too high) has occurred while \output is active [] [38] Underfull \hbox (badness 10000) in paragraph at lines 3357--3358 [] Underfull \hbox (badness 10000) in paragraph at lines 3363--3370 [] Underfull \hbox (badness 10000) in paragraph at lines 3371--3382 [] Underfull \hbox (badness 10000) in paragraph at lines 3384--3387 [] Underfull \hbox (badness 10000) in paragraph at lines 3412--3419 [] [39] Underfull \vbox (badness 10000) has occurred while \output is active [] Overfull \vbox (1.63055pt too high) has occurred while \output is active [] [40] [41] (C:/FPhandbook/hyperconvex.aux) LaTeX Warning: Label(s) may have changed. Rerun to get cross-references right. ) Here is how much of TeX's memory you used: 1955 strings out of 2873 20119 string characters out of 28518 13922 words of memory out of 73728 4618 multiletter control sequences out of 10000 12821 words of font info for 52 fonts, out of 20000 for 255 176 hyphenation exceptions out of 607 34i,15n,32p,286b,417s stack positions out of 200i,40n,60p,5000b,600s Output written on C:\FPhandbook\hyperconvex.dvi (41 pages, 215136 bytes).