This is TeX, Version 3.14159 (preloaded format=LaTeX 99.4.14) 27 JUN 1999 11:46 **&LaTeX elementary-gates-9503016.tex (elementary-gates-9503016.tex LaTeX2e <1996/12/01> (:TeX-inputs:LaTeX:article.cls Document Class: article 1996/10/31 v1.3u Standard LaTeX document class (:TeX-inputs:LaTeX:size12.clo File: size12.clo 1996/10/31 v1.3u 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 ) (elementary-gates-9503016.aux) LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 110. LaTeX Font Info: ... okay on input line 110. LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 110. LaTeX Font Info: ... okay on input line 110. LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 110. LaTeX Font Info: ... okay on input line 110. LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 110. LaTeX Font Info: ... okay on input line 110. LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 110. LaTeX Font Info: ... okay on input line 110. LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 110. LaTeX Font Info: ... okay on input line 110. LaTeX Font Info: External font `cmex10' loaded for size (Font) <14.4> on input line 112. LaTeX Font Info: External font `cmex10' loaded for size (Font) <7> on input line 112. LaTeX Font Info: External font `cmex10' loaded for size (Font) <5> on input line 112. LaTeX Font Info: External font `cmex10' loaded for size (Font) <10.95> on input line 112. LaTeX Font Info: External font `cmex10' loaded for size (Font) <8> on input line 112. LaTeX Font Info: External font `cmex10' loaded for size (Font) <6> on input line 112. LaTeX Font Warning: Command \sl invalid in math mode on input line 112. [1 ] [2] LaTeX Font Info: External font `cmex10' loaded for size (Font) <12> on input line 160. [3] LaTeX Font Warning: Command \scriptsize invalid in math mode on input line 195. [4] [5] [6] [7] LaTeX Font Warning: Command \scriptsize invalid in math mode on input line 380. LaTeX Font Warning: Command \scriptsize invalid in math mode on input line 393. LaTeX Font Warning: Command \scriptsize invalid in math mode on input line 402. [8] LaTeX Font Warning: Command \scriptsize invalid in math mode on input line 421. LaTeX Font Warning: Command \scriptsize invalid in math mode on input line 429. [9] [10] ! LaTeX Error: Command \Box not provided in base LaTeX2e. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.602 ...absorbed into the second one.\qed \ee ? r OK, entering \nonstopmode... LaTeX Font Info: Try loading font information for OMS+cmr on input line 611. (:TeX-inputs:LaTeX:omscmr.fd File: omscmr.fd 1997/01/14 v2.5a Standard LaTeX font definitions ) LaTeX Font Info: Font shape `OMS/cmr/m/n' in size <12> not available (Font) Font shape `OMS/cmsy/m/n' tried instead on input line 611. [11] ! LaTeX Error: Command \Box not provided in base LaTeX2e. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.728 \qed \ee Load the latexsym or the amsfonts package to define this symbol [12] ! LaTeX Error: Command \Box not provided in base LaTeX2e. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.799 also be special unitary.\qed \ee Load the latexsym or the amsfonts package to define this symbol [13] ! LaTeX Error: Command \Box not provided in base LaTeX2e. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.855 \right).$$\qed \ee Load the latexsym or the amsfonts package to define this symbol [14] LaTeX Font Warning: Command \scriptsize invalid in math mode on input line 960. LaTeX Font Warning: Command \scriptsize invalid in math mode on input line 962. LaTeX Font Warning: Command \scriptsize invalid in math mode on input line 965. LaTeX Font Warning: Command \scriptsize invalid in math mode on input line 967. LaTeX Font Warning: Command \scriptsize invalid in math mode on input line 972. LaTeX Font Warning: Command \scriptsize invalid in math mode on input line 973. LaTeX Font Warning: Command \scriptsize invalid in math mode on input line 981. [15] ! LaTeX Error: Command \Box not provided in base LaTeX2e. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.988 as required.\qed \ee Load the latexsym or the amsfonts package to define this symbol ! LaTeX Error: Command \Box not provided in base LaTeX2e. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.1057 ...wedge_1(W \cdot \sigma_x)$).\qed \ee Load the latexsym or the amsfonts package to define this symbol [16] LaTeX Font Warning: Command \scriptsize invalid in math mode on input line 1204 . [17] LaTeX Font Warning: Command \scriptsize invalid in math mode on input line 1213 . ! LaTeX Error: Command \Box not provided in base LaTeX2e. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.1215 the third is $V \cdot V = U$.\qed \ee Load the latexsym or the amsfonts package to define this symbol LaTeX Font Warning: Command \scriptsize invalid in math mode on input line 1221 . LaTeX Font Warning: Command \scriptsize invalid in math mode on input line 1237 . LaTeX Font Warning: Command \scriptsize invalid in math mode on input line 1237 . LaTeX Font Warning: Command \scriptsize invalid in math mode on input line 1239 . [18] [19] LaTeX Font Warning: Command \scriptsize invalid in math mode on input line 1328 . LaTeX Font Warning: Command \scriptsize invalid in math mode on input line 1329 . LaTeX Font Warning: Command \scriptsize invalid in math mode on input line 1397 . LaTeX Font Warning: Command \scriptsize invalid in math mode on input line 1399 . LaTeX Font Info: External font `cmex10' loaded for size (Font) <17.28> on input line 1405. [20] LaTeX Font Warning: Command \scriptsize invalid in math mode on input line 1487 . LaTeX Font Warning: Command \scriptsize invalid in math mode on input line 1489 . LaTeX Font Warning: Command \scriptsize invalid in math mode on input line 1491 . LaTeX Font Warning: Command \scriptsize invalid in math mode on input line 1504 . LaTeX Font Warning: Command \scriptsize invalid in math mode on input line 1506 . LaTeX Font Warning: Command \scriptsize invalid in math mode on input line 1508 . LaTeX Font Warning: Command \scriptsize invalid in math mode on input line 1514 . LaTeX Font Warning: Command \scriptsize invalid in math mode on input line 1519 . LaTeX Font Warning: Command \scriptsize invalid in math mode on input line 1535 . [21] Overfull \hbox (42.70886pt too wide) in alignment at lines 1542--1551 [][][] [] [] [22] ! LaTeX Error: Command \Box not provided in base LaTeX2e. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.1690 bits.\qed \ee Load the latexsym or the amsfonts package to define this symbol [23] ! LaTeX Error: Command \Box not provided in base LaTeX2e. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.1784 By inspection.\qed \ee Load the latexsym or the amsfonts package to define this symbol ! LaTeX Error: Command \Box not provided in base LaTeX2e. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.1807 \qed \ee Load the latexsym or the amsfonts package to define this symbol [24] LaTeX Font Warning: Command \scriptsize invalid in math mode on input line 1908 . [25] ! LaTeX Error: Command \Box not provided in base LaTeX2e. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.1916 setting $V$ so that $V^2 = U$.\qed \ee Load the latexsym or the amsfonts package to define this symbol LaTeX Font Warning: Command \scriptsize invalid in math mode on input line 1927 . ! LaTeX Error: Command \Box not provided in base LaTeX2e. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.1935 ...t $C_{n-1} \in \Theta(n^2)$.\qed \ee Load the latexsym or the amsfonts package to define this symbol [26] ! LaTeX Error: Command \Box not provided in base LaTeX2e. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.1967 ...t compute $\wedge_{n-1}(U)$.\qed \ee Load the latexsym or the amsfonts package to define this symbol [27] LaTeX Font Warning: Command \scriptsize invalid in math mode on input line 2004 . LaTeX Font Warning: Command \scriptsize invalid in math mode on input line 2014 . LaTeX Font Warning: Command \scriptsize invalid in math mode on input line 2017 . LaTeX Font Warning: Command \scriptsize invalid in math mode on input line 2029 . LaTeX Font Warning: Command \scriptsize invalid in math mode on input line 2030 . LaTeX Font Warning: Command \scriptsize invalid in math mode on input line 2031 . [28] ! LaTeX Error: Command \Box not provided in base LaTeX2e. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.2043 ...wedge_{n-1}(U)$ within $\e$.\qed \ee Load the latexsym or the amsfonts package to define this symbol ! LaTeX Error: Command \Box not provided in base LaTeX2e. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.2138 Lemma 4.3.\qed \ee Load the latexsym or the amsfonts package to define this symbol [29] [30] ! LaTeX Error: Command \Box not provided in base LaTeX2e. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.2245 By inspection.\qed \ee Load the latexsym or the amsfonts package to define this symbol [31] [32] [33] [34] [35] [36] LaTeX Font Warning: Command \scriptsize invalid in math mode on input line 2597 . [37] [38] (elementary-gates-9503016.aux) ) Here is how much of TeX's memory you used: 367 strings out of 5929 3618 string characters out of 28856 50848 words of memory out of 200001 3243 multiletter control sequences out of 10000 12398 words of font info for 42 fonts, out of 60000 for 256 14 hyphenation exceptions out of 2551 30i,13n,21p,219b,333s stack positions out of 600i,150n,200p,3000b,2000s Output written on elementary-gates-9503016.dvi (38 pages, 101064 bytes).