%   charset.tex January 11 1993
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%   This file is adapted from the file manend.tex 
%   Version 3.0 of REVTeX, November 10, 1992.
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\documentstyle[aps]{revtex}

\def\btt#1{{\tt$\backslash$\string#1}}%

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\def\slantfrac#1#2{\kern1em^{#1}\kern-.3em/\kern-.1em_{#2}}

\begin{document}

%\appendix
%\twocolumn
\narrowtext

\def\xxx{4pt}


\subsubsection[Standard \protect\LaTeX{} symbols]{Standard LaTeX symbols}
The following tables show the standard symbols for \LaTeX{} users.

\begin{table}
\caption{Text accents with letter a.}
\begin{tabular}{c@{\hspace{\xxx}}lc@{\hspace{\xxx}}lc@{\hspace{\xxx}}%
                lc@{\hspace{\xxx}}l}
\`{a} & \verb+\`{a}+&
\'{a} & \verb+\'{a}+&
\^{a} & \verb+\^{a}+&
\"{a} & \verb+\"{a}+\\
\~{a} & \verb+\~{a}+&
\={a} & \verb+\={a}+&
\.{a} & \verb+\.{a}+&
\u{a} & \verb+\u{a}+\\
\v{a} & \verb+\v{a}+&
\H{a} & \verb+\H{a}+&
\t{aa} & \verb+\t{aa}+&
\c{a} & \verb+\c{a}+\\
\d{a} & \verb+\d{a}+&
\b{a} & \verb+\b{a}+
\end{tabular}
\end{table}


\begin{table}
\caption{Math accents with letter a.}
\begin{tabular}%
  {c@{\hspace{\xxx}}lc@{\hspace{\xxx}}lc@{\hspace{\xxx}}lc@{\hspace{\xxx}}l}
$\hat{a}$ & \verb+\hat{a}+ &
$\check{a}$ & \verb+\check{a}+ &
 $\dot{a}$ & \verb+\dot{a}+ &
 $\ddot{a}$ & \verb+\ddot{a}+ \\
$\breve{a}$ & \verb+\breve{a}+ &
 $\tilde{a}$ & \verb+\tilde{a}+ &
 $\grave{a}$ & \verb+\grave{a}+ &
 $\acute{a}$ & \verb+\acute{a}+ \\
  $\bar{a}$ & \verb+\bar{a}+ &
  $\vec{a}$ & \verb+\vec{a}+ &
\end{tabular}
\end{table}

\newpage

\begin{table}
\caption{Special symbols; any mode.}
\begin{tabular}{c@{\hspace{\xxx}}lc@{\hspace{\xxx}}lc@{\hspace{\xxx}}l}
\dag & \verb+\dag+& \S & \verb+\S+& \copyright & \verb+\copyright+\\
\ddag & \verb+\ddag+& \P & \verb+\P+& \pounds & \verb+\pounds+\\
\end{tabular}
\end{table}




\begin{table}
\caption{Other special (foreign) symbols; text mode.}
\begin{tabular}%
  {c@{\hspace{\xxx}}lc@{\hspace{\xxx}}lc@{\hspace{\xxx}}lc@{\hspace{\xxx}}l}
 \aa & \verb+\aa+&
 \AA & \verb+\AA+&
\ae & \verb+\ae+&
\AE & \verb+\AE+\\
 \o & \verb+\o+&
 \O & \verb+\O+&
\oe & \verb+\oe+&
\OE & \verb+\OE+\\
 \l & \verb+\l+&
 \L & \verb+\L+&
 ?` & \verb+?`+&
 !` & \verb+!`+\\
 \ss & \verb+\ss+
\end{tabular}
\end{table}


\newpage

\begin{table}
\caption{Greek letters; used in math mode.}
\begin{tabular}%
  {c@{\hspace{\xxx}}lc@{\hspace{\xxx}}lc@{\hspace{\xxx}}lc@{\hspace{\xxx}}l}
\multicolumn{8}{c}{\it Lowercase}\\
$\alpha$ & \verb+\alpha+ &
$\beta$ & \verb+\beta+ &
$\gamma$ & \verb+\gamma+ &
$\delta$ & \verb+\delta+\\
$\epsilon$ & \verb+\epsilon+ &
$\varepsilon$ & \verb+\varepsilon+ &
$\zeta$ & \verb+\zeta+ &
$\eta$ & \verb+\eta+ \\
 $\theta$ & \verb+\theta+ &
 $\vartheta$ & \verb+\vartheta+ &
 $\iota$ & \verb+\iota+ &
 $\kappa$ & \verb+\kappa+ \\
 $\lambda$ & \verb+\lambda+ &
 $\mu$ & \verb+\mu+ &
 $\nu$ & \verb+\nu+ &
 $\xi$ & \verb+\xi+\\
 $o$ & \verb+o+ &
 $\pi$ & \verb+\pi+ &
 $\varpi$ & \verb+\varpi+ &
 $\rho$ & \verb+\rho+ \\
   $\varrho$ & \verb+\varrho+ &
   $\sigma$ & \verb+\sigma+ &
 $\varsigma$ & \verb+\varsigma+ &
    $\tau$ & \verb+\tau+ \\
   $\upsilon$ & \verb+\upsilon+ &
   $\phi$ & \verb+\phi+ &
   $\varphi$ & \verb+\varphi+ &
 $\chi$ & \verb+\chi+ \\
 $\psi$ & \verb+\psi+ &
  $\omega$ & \verb+\omega+ &
 \\[\baselineskip]
\multicolumn{8}{c}{\it Uppercase}\\
$\Gamma$ & \verb+\Gamma+ &
$\Delta$ & \verb+\Delta+ &
$\Theta$ & \verb+\Theta+ &
 $\Lambda$ & \verb+\Lambda+\\
 $\Xi$ & \verb+\Xi+ &
 $\Pi$ & \verb+\Pi+ &
   $\Sigma$ & \verb+\Sigma+ &
    $\Upsilon$ & \verb+\Upsilon+ \\
 $\Phi$ & \verb+\Phi+ &
 $\Psi$ & \verb+\Psi+ &
 $\Omega$ & \verb+\Omega+ &
\end{tabular}
\end{table}

\newpage

\begin{table}
\caption{Binary operation symbols; used in math mode.}
\def\xxx{3.2pt}
\begin{tabular}%
  {c@{\hspace{\xxx}}lc@{\hspace{\xxx}}lc@{\hspace{\xxx}}lc@{\hspace{\xxx}}l}
$\pm$ & \verb+\pm+ &
$\mp$ & \verb+\mp+ &
$\times$ & \verb+\times+ &
$\div$ & \verb+\div+ \\
$\ast$ & \verb+\ast+ &
$\star$ & \verb+\star+ &
$\circ$ & \verb+\circ+ &
$\bullet$ & \verb+\bullet+\\
 $\cap$ & \verb+\cap+ &
 $\cup$ & \verb+\cup+ &
 $\uplus$ & \verb+\uplus+ &
$\cdot$ & \verb+\cdot+ \\
 $\sqcap$ & \verb+\sqcap+  &
 $\sqcup$ & \verb+\sqcup+ &
 $\vee$ & \verb+\vee+ &
 $\wedge$ & \verb+\wedge+\\
   $\oplus$ & \verb+\oplus+ &
 $\ominus$ & \verb+\ominus+ &
 $\otimes$ & \verb+\otimes+ &
 $\oslash$ & \verb+\oslash+ \\
$\bigtriangleup$ & \verb+\bigtriangleup+ &
 $\odot$ & \verb+\odot+ &
  $\lhd$ & \verb+\lhd+ &
 $\dagger$ & \verb+\dagger+ \\
$\bigtriangledown$ & \verb+\bigtriangledown+ &
 $\bigcirc$ & \verb+\bigcirc+ &
  $\rhd$ & \verb+\rhd+ &
 $\ddagger$ & \verb+\ddagger+\\
$\triangleleft$ & \verb+\triangleleft+ &
 $\diamond$ & \verb+\diamond+ &
  $\unlhd$ & \verb+\unlhd+ &
 $\setminus$ & \verb+\setminus+\\
$\triangleright$ & \verb+\triangleright+ &
 $\wr$ & \verb+\wr+  &
 $\unrhd$ & \verb+\unrhd+  &
  $\amalg$ & \verb+\amalg+
\end{tabular}
\end{table}


\newpage

\begin{table}
\caption{Relation symbols; used in math mode. }
\begin{tabular}%
  {c@{\hspace{\xxx}}lc@{\hspace{\xxx}}lc@{\hspace{\xxx}}lc@{\hspace{\xxx}}l}
$\leq$ & \verb+\leq+ &
 $\geq$ & \verb+\geq+ &
$\ll$ & \verb+\ll+ &
 $\gg$ & \verb+\gg+ \\
 $\equiv$ & \verb+\equiv+ &
 $\asymp$ & \verb+\asymp+ &
   $\neq$ & \verb+\neq+ &
   $\doteq$ & \verb+\doteq+  \\
$\subset$ & \verb+\subset+ &
 $\supset$ & \verb+\supset+ &
$\subseteq$ & \verb+\subseteq+ &
 $\supseteq$ & \verb+\supseteq+  \\
$\sqsubset$ & \verb+\sqsubset+ &
 $\sqsupset$ & \verb+\sqsupset+ &
$\sqsubseteq$ & \verb+\sqsubseteq+ &
 $\sqsupseteq$ & \verb+\sqsupseteq+\hidewidth \\
   $\models$ & \verb+\models+ &
   $\perp$ & \verb+\perp+  &
 $\mid$ & \verb+\mid+  &
   $\parallel$ & \verb+\parallel+ \\
$\prec$ & \verb+\prec+ &
 $\succ$ & \verb+\succ+ &
$\preceq$ & \verb+\preceq+ &
 $\succeq$ & \verb+\succeq+ \\
 $\sim$ & \verb+\sim+ &
   $\simeq$ & \verb+\simeq+ &
   $\approx$ & \verb+\approx+ &
   $\cong$ & \verb+\cong+ \\
 $\bowtie$ & \verb+\bowtie+  &
 $\Join$ & \verb+\Join+  &
 $\smile$ & \verb+\smile+ &
 $\frown$ & \verb+\frown+ \\
$\in$ & \verb+\in+ &
 $\ni$ & \verb+\ni+ &
$\vdash$ & \verb+\vdash+ &
 $\dashv$ & \verb+\dashv+  \\
 $\propto$ & \verb+\propto+ &
\end{tabular}
%\end{table}

 Negated relations can sometimes
be constructed with \verb+\not+. For example,
\[
\hbox{\verb+If $x \not< y$ then $x \not\leq z$.+}
\]
gives
\[
\hbox{If $x \not< y$ then $x \not\leq z$.}
\]

\end{table}

\newpage

\begin{table}
\caption{Arrow symbols; used in math mode.}
\begin{tabular}{c@{\hspace{\xxx}}lc@{\hspace{\xxx}}l}
$\leftarrow$ & \verb+\leftarrow+ &
$\rightarrow$ & \verb+\rightarrow+ \\
 $\longleftarrow$ & \verb+\longleftarrow+ &
 $\longrightarrow$ & \verb+\longrightarrow+ \\
$\Leftarrow$ & \verb+\Leftarrow+ &
$\Rightarrow$ & \verb+\Rightarrow+ \\
 $\Longleftarrow$ & \verb+\Longleftarrow+ &
 $\Longrightarrow$ & \verb+\Longrightarrow+ \\
$\hookleftarrow$ & \verb+\hookleftarrow+ & $\hookrightarrow$
  & \verb+\hookrightarrow+ \\
$\leftharpoonup$ & \verb+\leftharpoonup+ & $\rightharpoonup$
  & \verb+\rightharpoonup+\\
$\leftharpoondown$ & \verb+\leftharpoondown+ & $\rightharpoondown$
  & \verb+\rightharpoondown+ \\
$\rightleftharpoons$&\verb+\rightleftharpoons+&$\leadsto$&\verb+\leadsto+ \\
$\leftrightarrow$ & \verb+\leftrightarrow+ & $\longleftrightarrow$
  & \verb+\longleftrightarrow+  \\
$\Leftrightarrow$ & \verb+\Leftrightarrow+ & $\Longleftrightarrow$
  & \verb+\Longleftrightarrow+  \\
$\mapsto$ & \verb+\mapsto+ & $\longmapsto$ & \verb+\longmapsto+ \\
\multicolumn{4}{c}{%
   \begin{tabular}{c@{\hspace{\xxx}}l}
   $\uparrow$ & \btt{uparrow}\\
   $\downarrow$ & \btt{downarrow} \\
   $\Uparrow$ & \btt{Uparrow}\\
   $\Downarrow$ & \btt{Downarrow}\\
   $\updownarrow$ & \btt{updownarrow}\\
   $\Updownarrow$ & \btt{Updownarrow}\\
   $\nearrow$ & \btt{nearrow}\\
   $\searrow$ & \btt{searrow}\\
   $\swarrow$ & \btt{swarrow}\\
   $\nwarrow$ & \btt{nwarrow}
   \end{tabular}%
} % end of multicolumn
%
\end{tabular}
\end{table}



\newpage

\begin{table}
\caption{Miscellaneous symbols; used in math mode.}
\begin{tabular}%
  {c@{\hspace{\xxx}}lc@{\hspace{\xxx}}lc@{\hspace{\xxx}}lc@{\hspace{\xxx}}l}
 $\flat$ & \verb+\flat+ &
 $\natural$ & \verb+\natural+ &
 $\sharp$ & \verb+\sharp+ &
 $\prime$ & \verb+\prime+ \\
 $\backslash$ & \verb+\backslash+ &
 $\forall$ & \verb+\forall+ &
  $\infty$ & \verb+\infty+ &
 $\exists$ & \verb+\exists+\\
 $\emptyset$ & \verb+\emptyset+ &
  $\Box$ & \verb+\Box+ &
 $\nabla$ & \verb+\nabla+ &
 $\neg$ & \verb+\neg+ \\
  $\Diamond$ & \verb+\Diamond+ &
 $\surd$ & \verb+\surd+ &
  $\triangle$ & \verb+\triangle+&
 $\|$ & \verb+\|+ \\
  $\clubsuit$ & \verb+\clubsuit+ &
$\aleph$ & \verb+\aleph+ &
$\wp$ & \verb+\wp+ &
 $\top$ & \verb+\top+\\
  $\diamondsuit$ & \verb+\diamondsuit+&
$\Re$ & \verb+\Re+ &
$\ell$ & \verb+\ell+ &
 $\bot$ & \verb+\bot+ \\
  $\heartsuit$ & \verb+\heartsuit+ &
$\Im$ & \verb+\Im+ &
$\imath$ & \verb+\imath+ &
 $\partial$ & \verb+\partial+ \\
  $\spadesuit$ & \verb+\spadesuit+ &
$\hbar$ & \verb+\hbar+ &
$\jmath$ & \verb+\jmath+ &
 $\angle$ & \verb+\angle+ \\
$\mho$ & \verb+\mho+
\end{tabular}
\end{table}

\newpage

\begin{table}
\caption{Log-like functions; used in math mode.}
\begin{tabular}{llllllll}
\verb+\arccos+ &
\verb+\arcsin+ &
\verb+\arctan+ &
\verb+\arg+ &
 \verb+\cos+ \\
 \verb+\cosh+ &
 \verb+\cot+ &
 \verb+\coth+ &
 \verb+\csc+ &
 \verb+\deg+ \\
 \verb+\det+ &
 \verb+\dim+ &
 \verb+\exp+ &
 \verb+\gcd+ &
 \verb+\hom+ \\
 \verb+\inf+ &
 \verb+\ker+ &
 \verb+\lg+ &
 \verb+\lim+ &
 \verb+\liminf+ \\
  \verb+\limsup+ &
  \verb+\ln+ &
  \verb+\log+ &
  \verb+\max+ &
 \verb+\min+ \\
 \verb+\Pr+ &
 \verb+\sec+ &
 \verb+\sin+ &
 \verb+\sinh+ &
 \verb+\sup+ \\
 \verb+\tan+ &
 \verb+\tanh+
\end{tabular}
\end{table}

\newpage

\begin{table}
\caption{Delimiters; used in math mode.}
\begin{tabular}{c@{\hspace{\xxx}}lc@{\hspace{\xxx}}lc@{\hspace{\xxx}}l}
$($ & \verb+(+ &
 $)$ & \verb+)+ &
$/$ & \verb+/+ \\
$[$ & \verb+[+ &
 $]$ & \verb+]+ &
 $\backslash$ & \verb+\backslash+ \\
$\{$ & \verb+\{+ &
 $\}$ & \verb+\}+ &
$|$ & \verb+|+ \\
$\langle$ & \verb+\langle+ &
 $\rangle$ & \verb+\rangle+ &
 $\|$ & \verb+\|+ \\
 $\uparrow$ & \verb+\uparrow+ &
  $\Uparrow$ & \verb+\Uparrow+&
$\lfloor$ & \verb+\lfloor+ \\
 $\downarrow$ & \verb+\downarrow+ &
  $\Downarrow$ & \verb+\Downarrow+ &
 $\rfloor$ & \verb+\rfloor+ \\
 $\updownarrow$ & \verb+\updownarrow+ &
  $\Updownarrow$ & \verb+\Updownarrow+ &
$\lceil$ & \verb+\lceil+ \\
 & & & & $\rceil$ & \verb+\rceil+
\end{tabular}
\end{table}


\newpage

\begin{table}
\caption{Miscellaneous symbols; used in math mode.}
\begin{tabular}{c@{\hspace{\xxx}}lc@{\hspace{\xxx}}lc@{\hspace{\xxx}}l}
$\textstyle\sum$ $\displaystyle\sum$ & \verb+\sum+ &
$\textstyle\prod$ $\displaystyle\prod$ & \verb+\prod+ &
$\textstyle\coprod$ $\displaystyle\coprod$ & \verb+\coprod+ \\
$\textstyle\int$ $\displaystyle\int$ & \verb+\int+ &
$\textstyle\oint$ $\displaystyle\oint$ & \verb+\oint+ &
$\textstyle\biguplus$ $\displaystyle\biguplus$ & \verb+\biguplus+ \\
  $\textstyle\bigcap$ $\displaystyle\bigcap$ & \verb+\bigcap+ &
  $\textstyle\bigcup$ $\displaystyle\bigcup$ & \verb+\bigcup+ &
  $\textstyle\bigsqcup$ $\displaystyle\bigsqcup$ & \verb+\bigsqcup+ \\
  $\textstyle\bigodot$ $\displaystyle\bigodot$ & \verb+\bigodot+ &
  $\textstyle\bigotimes$ $\displaystyle\bigotimes$ & \verb+\bigotimes+ &
  $\textstyle\bigoplus$ $\displaystyle\bigoplus$ & \verb+\bigoplus+ \\
  $\textstyle\bigvee$ $\displaystyle\bigvee$ & \verb+\bigvee+ &
  $\textstyle\bigwedge$ $\displaystyle\bigwedge$ & \verb+\bigwedge+
\end{tabular}
\end{table}

\newpage

\subsubsection[Standard \protect\LaTeX{} typefaces]{Standard LaTeX typefaces}

You can access a pair of special typefaces in \LaTeX.

% only math mode?
You can switch to script (calligraphic) letters by using the \verb+\cal+
command (note the $\cal L$):
\begin{verbatim}
{\cal L}_{\text{int}} = e F^{3}_{\pi} r^{2}
  B^{0}(r,t) \epsilon \sin(\Omega t)
  \exp(\eta t),
\end{verbatim}
gives
\[
{\cal L}_{\text{int}} = eF^{3}_{\pi} r^{2}
 B^{0}(r,t)\epsilon\sin(\Omega t)\exp(\eta t),
\]
Only uppercase letters are available in the \verb+\cal+ font.


% only math mode?
You can switch to sans serif letters by using the \verb+\sf+
command (note the $\sf M$):

% \case -> \frac for portability

\begin{verbatim}
R({\cal Q}-{\cal Q}_{0})
=R_{0} \exp\left(-\frac{1}{2}\Delta {\cal Q}
 \cdot{\sf M}\cdot\Delta{\cal Q}\right).
  \label{eq:rdef}
\end{verbatim}
gives
\[
  R({\cal Q}-{\cal Q}_{0}) =
%  R_{0} \exp\left(-\case1/2\Delta {\cal Q} \cdot {\sf M}
  R_{0} \exp\left(-\frac{1}{2}\Delta {\cal Q} \cdot {\sf M}
  \cdot \Delta {\cal Q}\right).
\]
Both uppercase and lowercase letters are available with \verb+\sf+.

\newpage

\subsubsection{Other notations}

% math mode only?
The \verb+\overline+ command puts a horizontal line above its argument
in math mode:
\begin{verbatim}
$\overline{x}+\overline{y}$
\end{verbatim}
gives
\[
\overline{x}+\overline{y}
\]

There is an analogous \verb+\underline+ command that works in text
or math mode:
\begin{verbatim}
The equation \underline{is} $\underline{x+y}$.
\end{verbatim}
gives
\[
\hbox{The equation \underline{is} $\underline{x+y}$.}
\]

% math mode?
Horizontal braces are put above or below an expression with the
\verb+\overbrace+ and \verb+\underbrace+ commands:
\begin{verbatim}
$\underbrace{a_{1} + \overbrace{a_{2}+a_{3}}
  + a_{4}}$
\end{verbatim}
gives
\[
\underbrace{a_{1} + \overbrace{a_{2}+a_{3}} + a_{4}}
\]
and in displayed math, a subscript or a superscript puts a label on
the brace:
\begin{verbatim}
$\underbrace{a_{1} +
  \overbrace{a_{2}+\cdots+a_{n-1}}^{n-2}
   + a_{n}}_{n}$
\end{verbatim}
gives
\[
\underbrace{a_{1} + \overbrace{a_{2}+\cdots+a_{n-1}}^{n-2} + a_{n}}_{n}
\]

Wide versions of the \verb+\hat+ and \verb+\tilde+ commands are available.
They are called \verb+\widehat+ and \verb+\widetilde+, respectively.
Here is an example:
\begin{verbatim}
$\widehat{a} + \widehat{ab}
 + \widehat{abc} + \widehat{abcd}$
\end{verbatim}
gives
\[
\widehat{a} + \widehat{ab} + \widehat{abc} + \widehat{abcd}
\]

\end{document}
% end of file charset.tex