%-*-tex-*- \ifundefined{writestatus} \input status \relax \fi % \chcode{matsym} \def\cqu{\cquote{{\leftskip=0ptplus1fill\def\\% {\endgraf}Yet what are all such gaieties to me\\Whose thoughts are full of indices and surds?\\$x^2+7x+53$\\ $\displaystyle={11\over3}$.\\ }}{ Four~Riddles, Lewis~Carroll (1832-1898)}} \chapterhead{matsym}{MATHEMATICS:\cr SYMBOLS\cr FONTS} \tex\ supplies a vast array of mathematical symbols, both in fixed and extensible forms. For the most part, the symbols mentioned are available only in mathematics mode. Other symbols that are useful in regular text mode have been listed in Section \ref{specialtextsymbols} This chapter describes the symbols available. These symbols can be used in any font size that is on the system. Although it is possible to build your own symbols from simpler ones, it takes a little care to make sure that they are the correct size for all possible uses. \shead{mathfonts}{Ordinary Fonts in Mathematics} The mathematics symbol font contains characters that are very similar to to text italic font. The major difference is the spacing. {\it For instance} is in ordinary italic but $For\ instance$, is in math italic and was invoked with |$For\|\]|instance$|, where |\|\] forces the space. However, it is an easy matter to put roman or {\bf bold} letters in mathematics by just using |$x={\bf the\ spot}$| to obtain $x={\bf the\ spot}$. The real fun comes with something like the following: \def\xbf{{\vec{\bf x}}} $$ \xbf = 2^{\xbf^\xbf} $$ which is given by \begintt \def\xbf{{\vec{\bf x}}} $$ \xbf = 2^{\xbf^\xbf} $$ \endtt Note that the bold face is scaled correctly up into the superscripts. The extra $\{\ldots\}$ are necessary so that the third |\xbf| will look like a single token to the second |\@|. \shead{mathtypes}{Mathematical Symbol Types} Some symbols, even though they may look the same are given different command sequences in \tex. The reason for this is that their use demands different spacing and line breaking when in mathematics mode. For this reason there are four mathematics symbols for a vertical bar $\mid$, namely |\mid|, |\left\vert|, |\right\vert|, and |\vert|. The first is a {\it relation} -- called {\it Rel} -- like the $=$; the second and third are opening and closing brackets like ( and ) --- \tex\ calls these {\it openings} and {\it closings}; and the last is an {\it ordinary} --- called {\it Ord} --- symbol like $\infty$. Two other types are {\it large operators} --- called {\it Op} --- like $\sum$ and {\it binary operators} --- called {\it Bin} --- like $+$. Finally there is the {\it punctuation} --- called {\it Punct} --- like the period. \shead{greekletters}{Greek Letters} All the Greek letters are of type {\it Ord}. They are all assumed to be in mathematics mode. \bshortcomlist \@|\alpha {\rm A} A|&$\alpha$ A $A$ \cr \@|\beta {\rm B} B|&$\beta$ B $B$\cr \@|\gamma \Gamma {\mit\Gamma}|&$\gamma$ $\Gamma$ ${\mit\Gamma}$ \cr \@|\delta \Delta {\mit\Delta} |&$\delta$ $\Delta$ ${\mit\Delta}$ \cr \@|\epsilon {\rm E} E|&$\epsilon$ E $E$ \cr \@|\varepsilon|&$\varepsilon$\cr \@|\zeta {\rm Z} Z|&$\zeta$ Z $Z$ \cr \@|\eta {\rm H} H|&$\eta$ H $H$ \cr \@|\theta \Theta {\mit\Theta} |&$\theta$ $\Theta$ $\mit\Theta$ \cr \@|\vartheta|&$\vartheta$ \cr \@|\iota {\rm I} I|&$\iota$ I $I$ \cr \@|\kappa {\rm K} K |&$\kappa$ K $K$ \cr \@|\lambda \Lambda {\mit\Lambda} |&$\lambda$ $\Lambda$ $\mit\Lambda$ \cr \@|\mu {\rm M} M |&$\mu$ M $M$ \cr \@|\nu {\rm N} N|&$\nu$ N $N$ \cr \@|\xi \Xi {\mit\Xi} |&$\xi$ $\Xi$ $\mit\Xi$ \cr |o {\rm O} O|&$o$ O $O$ \cr \@|\pi \Pi {\mit\Pi}|&$\pi$ $\Pi$ $\mit\Pi$ \cr \@|\varpi|&$\varpi$\cr \@|\rho {\rm R} R|&$\rho$ R $R$\cr \@|\sigma \Sigma {\mit\Sigma}|&$\sigma$ $\Sigma$ $\mit\Sigma$ \cr \@|\tau {\rm T} T|&$\tau$ T $T$ \cr \@|\upsilon \Upsilon {\mit\Upsilon}|&$\upsilon$ $\Upsilon$ $\mit\Upsilon$ \cr \@|\phi \Phi {\mit\Phi}|&$\phi$ $\Phi$ $\mit\Phi$ \cr \@|\varphi|&$\varphi$ \cr \@|\chi {\rm X} X|&$\chi$ X $X$ \cr \@|\psi \Psi {\mit\Psi}|&$\psi$ $\Psi$ $\mit\Psi$\cr \@|\omega \Omega {\mit\Omega}|&$\omega$ $\Omega$ $\mit\Omega$\cr \eshortcomlist \shead{caligraphic}{Calligraphic Capitals} These look like ${\cal A} \ldots{\cal Z}$ and are obtained in math mode by the construction |{\cal }| while in math mode. \shead{opsrels}{Ords, Operators, and Relations} The following is a composite list of the symbols of type ordinary, relation and operator supplied by {\it plain}. The type is specified along with the symbol. The meaning of the type abbreviations has been given in Section \ref{mathtypes}. \medbreak \beginmulticolumnformat \hfuzz=20pt \parindent=0pt \def\bin{\leavevmode\hbox to 30pt{\it Bin\hss}} \def\ord{\leavevmode\hbox to 30pt{\it Ord\hss}} \def\rel{\leavevmode\hbox to 30pt{\it Rel\hss}} \def\op{\leavevmode\hbox to 30pt{\it Op\hss}} \intercolumnsep = {\hskip 3em} \numberofcolumns = 2 \obeylines \bin\hbox to 20pt{$+$\quad\hss}\@|+| \bin\hbox to 20pt{$-$\quad\hss}\@|-| \rel\hbox to 20pt{$=$\quad\hss}\@|=| \rel\hbox to 20pt{$\not=$\quad\hss}\@|\not=| |\ne| |\neq| \rel\hbox to 20pt{$<$\quad\hss}\@|<| \rel\hbox to 20pt{$\not<$\quad\hss}\@|\not<| \rel\hbox to 20pt{$>$\quad\hss}\@|>| \rel\hbox to 20pt{$\not>$\quad\hss}\@|\not>| \ord\hbox to 20pt{$\aleph$\quad\hss}\@|\aleph| \ord\hbox to 20pt{$\amalg$\quad\hss}\@|\amalg| \ord\hbox to 20pt{$\angle$\quad\hss}\@|\angle| \rel\hbox to 20pt{$\approx$\quad\hss}\@|\approx| \rel\hbox to 20pt{$\not\approx$\quad\hss}\@|\not\approx| \bin\hbox to 20pt{$\ast$\quad\hss}\@|\ast| or * \rel\hbox to 20pt{$\asymp$\quad\hss}\@|\asymp| \ord\hbox to 20pt{$\backslash$\quad\hss}\@|\backslash| \op\hbox to 20pt{$\bigcap$\quad\hss}\@|\bigcap| \bin\hbox to 20pt{$\bigcirc$\quad\hss}\@|\bigcirc| \op\hbox to 20pt{$\bigcup$\quad\hss}\@|\bigcup| \op\hbox to 20pt{$\bigodot$\quad\hss}\@|\bigodot| \op\hbox to 20pt{$\bigoplus$\quad\hss}\@|\bigoplus| \op\hbox to 20pt{$\bigotimes$\quad\hss}\@|\bigotimes| \op\hbox to 20pt{$\bigsqcup$\quad\hss}\@|\bigsqcup| \bin\hbox to 20pt{$\bigtriangledown$\quad\hss}\@|\bigtriangledown| \bin\hbox to 20pt{$\bigtriangleup$\quad\hss}\@|\bigtriangleup| \op\hbox to 20pt{$\biguplus$\quad\hss}\@|\biguplus| \op\hbox to 20pt{$\bigvee$\quad\hss}\@|\bigvee| \op\hbox to 20pt{$\bigwedge$\quad\hss}\@|\bigwedge| \ord\hbox to 20pt{$\bot$\quad\hss}\@|\bot| \rel\hbox to 20pt{$\bowtie$\quad\hss}\@|\bowtie| \bin\hbox to 20pt{$\bullet$\quad\hss}\@|\bullet| \bin\hbox to 20pt{$\cap$\quad\hss}\@|\cap| \bin\hbox to 20pt{$\cdot$\quad\hss}\@|\cdot| \bin\hbox to 20pt{$\circ$\quad\hss}\@|\circ| \ord\hbox to 20pt{$\clubsuit$\quad\hss}\@|\clubsuit| \rel\hbox to 20pt{$\cong$\quad\hss}\@|\cong| \rel\hbox to 20pt{$\not\cong$\quad\hss}\@|\not\cong| \op\hbox to 20pt{$\coprod$\quad\hss}\@|\coprod| \bin\hbox to 20pt{$\cup$\quad\hss}\@|\cup| \bin\hbox to 20pt{$\dagger$\quad\hss}\@|\dagger| \rel\hbox to 20pt{$\dashv$\quad\hss}\@|\dashv| \bin\hbox to 20pt{$\ddagger$\quad\hss}\@|\ddagger| \ord\hbox to 20pt{$\diamondsuit$\quad\hss}\@|\diamondsuit| \balancecolumnsize = 470pt \bin\hbox to 20pt{$\diamond$\quad\hss}\@|\diamond| \bin\hbox to 20pt{$\div$\quad\hss}\@|\div| \rel\hbox to 20pt{$\doteq$\quad\hss}\@|\doteq| \rel\hbox to 20pt{$\Downarrow$\quad\hss}\@|\Downarrow| \rel\hbox to 20pt{$\downarrow$\quad\hss}\@|\downarrow| \ord\hbox to 20pt{$\ell$\quad\hss}\@|\ell| \ord\hbox to 20pt{$\emptyset$\quad\hss}\@|\emptyset| \rel\hbox to 20pt{$\equiv$\quad\hss}\@|\equiv| \rel\hbox to 20pt{$\not\equiv$\quad\hss}\@|\not\equiv| \ord\hbox to 20pt{$\exists$\quad\hss}\@|\exists| \ord\hbox to 20pt{$\flat$\quad\hss}\@|\flat| \ord\hbox to 20pt{$\forall$\quad\hss}\@|\forall| \ord\hbox to 20pt{$\frown$\quad\hss}\@|\frown| \rel\hbox to 20pt{$\geq$\quad\hss}\@|\ge| \@|\geq| \rel\hbox to 20pt{$\not\geq$\quad\hss}\@|\not\geq| \rel\hbox to 20pt{$\gg$\quad\hss}\@|\gg| \rel\hbox to 20pt{$\not\gg$\quad\hss}\@|\not\gg| \ord\hbox to 20pt{$\hbar$\quad\hss}\@|\hbar| \ord\hbox to 20pt{$\heartsuit$\quad\hss}\@|\heartsuit| \rel\hbox to 20pt{$\hookleftarrow$\quad\hss}\@|\hookleftarrow| \rel\hbox to 20pt{$\hookrightarrow$\quad\hss}\@|\hookrightarrow| \ord\hbox to 20pt{$\Im$\quad\hss}\@|\Im| \ord\hbox to 20pt{$\imath$\quad\hss}\@|\imath| \ord\hbox to 20pt{$\infty$\quad\hss}\@|\infty| \op\hbox to 20pt{$\int$\quad\hss}\@|\int| \rel\hbox to 20pt{$\in$\quad\hss}\@|\in| \rel\hbox to 20pt{$\not\in$\quad\hss}\@|\not\in| \ord\hbox to 20pt{$\jmath$\quad\hss}\@|\jmath| \rel\hbox to 20pt{$\Leftarrow$\quad\hss}\@|\Leftarrow| \rel\hbox to 20pt{$\leftarrow$\quad\hss}\@|\leftarrow| \rel\hbox to 20pt{$\leftharpoondown$\quad\hss}\@|\leftharpoondown| \rel\hbox to 20pt{$\leftharpoonup$\quad\hss}\@|\leftharpoonup| \rel\hbox to 20pt{$\Leftrightarrow$\quad\hss}\@|\Leftrightarrow| \rel\hbox to 20pt{$\leftrightarrow$\quad\hss}\@|\leftrightarrow| \rel\hbox to 20pt{$\leq$\quad\hss}\@|\le| \@|\leq| \rel\hbox to 20pt{$\not\leq$\quad\hss}\@|\not\leq| \rel\hbox to 20pt{$\ll$\quad\hss}\@|\ll| \rel\hbox to 20pt{$\not\ll$\quad\hss}\@|\not\ll| \rel\hbox to 20pt{$\Longleftarrow$\quad\hss}\@|\Longleftarrow| \rel\hbox to 20pt{$\longleftarrow$\quad\hss}\@|\longleftarrow| \rel\hbox to 20pt{$\Longleftrightarrow$\quad\hss}\@|\Longleftrightarrow| \rel\hbox to 20pt{$\longleftrightarrow$\quad\hss}\@|\longleftrightarrow| \rel\hbox to 20pt{$\longmapsto$\quad\hss}\@|\longmapsto| \rel\hbox to 20pt{$\Longrightarrow$\quad\hss}\@|\Longrightarrow| \rel\hbox to 20pt{$\longrightarrow$\quad\hss}\@|\longrightarrow| \rel\hbox to 20pt{$\mapsto$\quad\hss}\@|\mapsto| \rel\hbox to 20pt{$\mid$\quad\hss}\@|\mid| \rel\hbox to 20pt{$\models$\quad\hss}\@|\models| \bin\hbox to 20pt{$\mp$\quad\hss}\@|\mp| \ord\hbox to 20pt{$\nabla$\quad\hss}\@|\nabla| \ord\hbox to 20pt{$\natural$\quad\hss}\@|\natural| \rel\hbox to 20pt{$\nearrow$\quad\hss}\@|\nearrow| \rel\hbox to 20pt{$\nwarrow$\quad\hss}\@|\nwarrow| \ord\hbox to 20pt{$\neg$\quad\hss}\@|\neg| \@|\lnot| \rel\hbox to 20pt{$\ni$\quad\hss}\@|\ni| \@|\owns| \bin\hbox to 20pt{$\odot$\quad\hss}\@|\odot| \op\hbox to 20pt{$\oint$\quad\hss}\@|\oint| \bin\hbox to 20pt{$\ominus$\quad\hss}\@|\ominus| \bin\hbox to 20pt{$\oplus$\quad\hss}\@|\oplus| \bin\hbox to 20pt{$\oslash$\quad\hss}\@|\oslash| \bin\hbox to 20pt{$\otimes$\quad\hss}\@|\otimes| \rel\hbox to 20pt{$\parallel$\quad\hss}\@|\parallel| \ord\hbox to 20pt{$\partial$\quad\hss}\@|\partial| \bin\hbox to 20pt{$\perp$\quad\hss}\@|\perp| \bin\hbox to 20pt{$\pm$\quad\hss}\@|\pm| \rel\hbox to 20pt{$\preceq$\quad\hss}\@|\preceq| \rel\hbox to 20pt{$\not\preceq$\quad\hss}\@|\not\preceq| \rel\hbox to 20pt{$\prec$\quad\hss}\@|\prec| \rel\hbox to 20pt{$\not\prec$\quad\hss}\@|\not\prec| \ord\hbox to 20pt{$\prime$\quad\hss}\@|\prime| \op\hbox to 20pt{$\prod$\quad\hss}\@|\prod| \rel\hbox to 20pt{$\propto$\quad\hss}\@|\propto| \ord\hbox to 20pt{$\Re$\quad\hss}\@|\Re| \rel\hbox to 20pt{$\Rightarrow$\quad\hss}\@|\Rightarrow| \rel\hbox to 20pt{$\rightarrow$\quad\hss}\@|\rightarrow| \rel\hbox to 20pt{$\rightharpoondown$\quad\hss}\@|\rightharpoondown| \rel\hbox to 20pt{$\rightharpoonup$\quad\hss}\@|\rightharpoonup| \rel\hbox to 20pt{$\rightleftharpoons$\quad\hss}\@|\rightleftharpoons| \bin\hbox to 20pt{$\setminus$\quad\hss}\@|\setminus| \ord\hbox to 20pt{$\sharp$\quad\hss}\@|\sharp| \rel\hbox to 20pt{$\simeq$\quad\hss}\@|\simeq| \rel\hbox to 20pt{$\not\simeq$\quad\hss}\@|\not\simeq| \rel\hbox to 20pt{$\sim$\quad\hss}\@|\sim| \rel\hbox to 20pt{$\not\sim$\quad\hss}\@|\not\sim| \rel\hbox to 20pt{$\smile$\quad\hss}\@|\smile| \ord\hbox to 20pt{$\spadesuit$\quad\hss}\@|\spadesuit| \bin\hbox to 20pt{$\sqcap$\quad\hss}\@|\sqcap| \bin\hbox to 20pt{$\sqcup$\quad\hss}\@|\sqcup| \rel\hbox to 20pt{$\sqsubseteq$\quad\hss}\@|\sqsubseteq| \rel\hbox to 20pt{$\not\sqsubseteq$\quad\hss}\@|\not\sqsubseteq| \rel\hbox to 20pt{$\sqsupseteq$\quad\hss}\@|\sqsupseteq| \rel\hbox to 20pt{$\not\sqsupseteq$\quad\hss}\@|\not\sqsupseteq| \ord\hbox to 20pt{$\star$\quad\hss}\@|\star| \rel\hbox to 20pt{$\subseteq$\quad\hss}\@|\subseteq| \rel\hbox to 20pt{$\not\subseteq$\quad\hss}\@|\not\subseteq| \rel\hbox to 20pt{$\subset$\quad\hss}\@|\subset| \rel\hbox to 20pt{$\not\subset$\quad\hss}\@|\not\subset| \rel\hbox to 20pt{$\succeq$\quad\hss}\@|\succeq| \rel\hbox to 20pt{$\not\succeq$\quad\hss}\@|\not\succeq| \rel\hbox to 20pt{$\succ$\quad\hss}\@|\succ| \rel\hbox to 20pt{$\not\succ$\quad\hss}\@|\not\succ| \op\hbox to 20pt{$\sum$\quad\hss}\@|\sum| \rel\hbox to 20pt{$\supseteq$\quad\hss}\@|\supseteq| \rel\hbox to 20pt{$\not\supseteq$\quad\hss}\@|\not\supseteq| \rel\hbox to 20pt{$\supset$\quad\hss}\@|\supset| \rel\hbox to 20pt{$\not\supset$\quad\hss}\@|\not\supset| \ord\hbox to 20pt{$\surd$\quad\hss}\@|\surd| \rel\hbox to 20pt{$\searrow$\quad\hss}\@|\searrow| \rel\hbox to 20pt{$\swarrow$\quad\hss}\@|\swarrow| \bin\hbox to 20pt{$\times$\quad\hss}\@|\times| \bin\hbox to 20pt{$\top$\quad\hss}\@|\top| \bin\hbox to 20pt{$\triangleleft$\quad\hss}\@|\triangleleft| \bin\hbox to 20pt{$\triangleright$\quad\hss}\@|\triangleright| \ord\hbox to 20pt{$\triangle$\quad\hss}\@|\triangle| \bin\hbox to 20pt{$\uplus$\quad\hss}\@|\uplus| \rel\hbox to 20pt{$\Uparrow$\quad\hss}\@|\Uparrow| \rel\hbox to 20pt{$\uparrow$\quad\hss}\@|\uparrow| \rel\hbox to 20pt{$\Updownarrow$\quad\hss}\@|\Updownarrow| \rel\hbox to 20pt{$\updownarrow$\quad\hss}\@|\updownarrow| \ord\hbox to 20pt{$\Vert$\quad\hss}\@|\Vert| or |\|\vrt \ord\hbox to 20pt{$\vert$\quad\hss}\@|\vert| or \vrt \rel\hbox to 20pt{$\vdash$\quad\hss}\@|\vdash| \bin\hbox to 20pt{$\vee$\quad\hss}\@|\lor| \@|\vee| \bin\hbox to 20pt{$\wedge$\quad\hss}\@|\land| \@|\wedge| \ord\hbox to 20pt{$\wp$\quad\hss}\@|\wp| \bin\hbox to 20pt{$\wr$\quad\hss}\@|\wr| \endmulticolumnformat \shead{delims}{Brackets or Delimiters} \tex\ supplies a large variety of delimiters or brackets, some of different sizes and others that grow to be the correct size. Here is a list of the delimiters supplied by {\it plain}. \bshortcomlist \@|\big\Arrowvert|¢er part of double up arrow -- $\big\Arrowvert$\cr \@|\big\arrowvert|¢er part of up arrow -- $\big\arrowvert$\cr \@|\langle|&left angle bracket -- $\langle$\cr \@|\rangle|&right angle bracket -- $\rangle$\cr \@|\big\bracevert|¢er part of large braces -- $\big\bracevert$ \cr |\{| or \@|\lbrace|&left brace -- $\{$\cr |\}| or \@|\rbrace|&right brace -- $\}$\cr |[| or \@|\lbrack|&left square bracket -- $[$\cr |]| or \@|\rbrack|&right square bracket -- $]$\cr \@|\lceil|&left ceiling -- $\lceil$\cr \@|\rceil|&right ceiling -- $\rceil$\cr \@|\lfloor|&left floor -- $\lfloor$\cr \@|\rfloor|&right floor -- $\rfloor$\cr \@|\big\lgroup|&large left paren without middle -- $\big\lgroup$\cr \@|\big\rgroup|&large right paren without middle -- $\big\rgroup$\cr \@|\big\lmoustache|& -- $\big\lmoustache$\cr \@|\big\rmoustache|& -- $\big\rmoustache$\cr |/|&slash -- $/$\cr \@|\backslash|&reverse slash -- $\backslash$\cr \vrt\ or |\vert|&vertical bar -- $\vert$\cr |\|\vrt\ or |\Vert|&double vertical bar -- $\Vert$\cr |\downarrow|&down arrow -- $\downarrow$\cr |\Downarrow|&double down arrow -- $\Downarrow$\cr |\uparrow|&up arrow -- $\uparrow$\cr |\Uparrow|&double up arrow -- $\Uparrow$\cr |\updownarrow|&up-and-down arrow -- $\updownarrow$\cr |\Updownarrow|&double up-and-down arrow -- $\Updownarrow$\cr |(|&left paren -- $($\cr |)|&right paren -- $)$\cr \eshortcomlist Two very powerful operators exist for the delimiters, namely |\left| and |\right|. They must be used in pairs as \par \hbox to \hsize{\hss\@|$\left ...\right$|,\hss} \noindent where the || is one of the above. The result of these operators is a pair of braces that grow the correct size for the particular formula in which they are used. This saves you from the problem of measuring. A |\right.| implies there is nothing at the place of the right delimiter and |\left.| similarly on the left. This allows a delimiter on only one side of a formula but to still have the operators match in pairs. Finally there is a second set of four pairs of operators, \@|\bigl|, |\Bigl|, \@|\biggl|, \@|\Biggl| and the corresponding right hand ones \@|\bigr|, \@|\Bigr|, \@|\biggr|, and \@|\Biggr|. These produce big, but fixed size delimiters for instance we have \begintt $$ \bigl\{\quad \Bigl[\quad \biggl\lfloor\quad \Biggl\Updownarrow\quad\Biggr\Updownarrow \quad\biggr\rfloor \quad\Bigr] \quad\bigr\} $$ \endtt produces $$ \bigl\{\quad \Bigl[\quad \biggl\lfloor\quad \Biggl\Updownarrow\quad\Biggr\Updownarrow \quad\biggr\rfloor \quad\Bigr] \quad\bigr\} $$ \shead{sincos}{Common Mathematical Functions} Certain functions such as |sin|, |cos|, and |limsup| for example are shown in mathematics text in Roman font. In addition some like |lim| sometimes take limits that should be placed in the same way as the limits on the large operators ({\it Op}). For example \begintt $$ \limsup_{t \leftrightarrow \infty} $$ \endtt gives $$ \limsup_{t \leftrightarrow \infty} $$ There are 32 of these already defined in {\it plain}. They are \beginthreecolumn \@|\arg| \@|\arccos| \@|\arcsin| \@|\arctan| \@|\cot| \@|\coth| \@|\cosh| \@|\cos| \@|\csc| \@|\det| \@|\deg| \@|\dim| \@|\exp| \@|\gcd| \@|\hom| \@|\inf| \@|\ker| \@|\lg| \@|\liminf| \@|\limsup| \@|\lim| \@|\ln| \@|\log| \@|\max| \@|\min| \@|\Pr| \@|\sec| \@|\sin| \@|\sinh| \@|\sup| \@|\tanh| \@|\tan| \endthreecolumn \shead{buildown}{Build Your Own Symbols} There are several advanced facilities in \tex\ for building your own symbols. However, there is one rather simple form using \@|\buildrel \over |. For example |\buildrel \triangle \over =| gives $\buildrel \triangle \over =$. \ejectpage \done