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\begin{document}

\title{\Huge{\bf Comprehensive Introduction to {\LaTeX} for Scientists and Engineers}}
\author{\huge{Hiro Shimoyama}}

\maketitle

\tableofcontents


\chapter{Introduction}
This manual was originally written several years ago because of my friends' request. 
The contents are arranged, so everyone can get started scientific theses or manuscripts 
without ``studying'' {\LaTeX}. It is not easy to read off the essence from some of {\LaTeX} books 
since the books usually contain every single detail. It could be a bit overwhelming if you do not know anything 
about {\LaTeX}. Some people seem to need a short introduction so they can 
explore by themselves later on. Therefore, this manual shows only the basics  
to get into {\LaTeX} by yourself. I am not explaining the philosophical or historical 
aspects on {\LaTeX} in this book. I tried to arrange this for physicists,  
mathematicians, chemists, and engineers so they can easily start writing a thesis or a manuscript. 
However, it is always a good idea to purchase a complete reference. There are so called 
bibles of {\TeX} written by initial developers.~\cite{Knuth,Lamport} These books might be a little old 
and you may want to buy some of newer ones which explain more commands and updated techniques. 

Here is the outline. In chapter \ref{ch:start}, it teaches how to set up a basic {\LaTeX}
environment. {\LaTeX} is the free software and you can use it on most of the operating systems.
In chapter \ref{ch:equation}, you will learn how to express mathematical equations.
In chapter \ref{ch:text}, this will introduce the basic techniques of how to format the text.
In chapter \ref{ch:figure}, this will explain how to manage pictures and figures.
Chapter \ref{ch:bib} will illustrate how to make a bibliography.
Chapter \ref{ch:ref} collects frequently-used symbols and commands so you can 
refer to them anytime.

\chapter{Starting Up \LaTeX} 		\label{ch:start}
\section{How to Download, Install, and Use \LaTeX}
If you are using Linux, you can go to the software center to find the related 
packages to install. You can also type \verb|latex| in the command 
prompt called ``terminal.'' Especially for Ubuntu Linux,\footnote{Linux is one of the 
computer operating systems based on UNIX. Ubuntu is one of them and it is close to 
Windows operating systems.} if you type that command in the terminal, the instruction 
will be given to install such as \verb|$ sudo apt-get install texlive|.\footnote{``texlive'' is an example.
Please follow the instruction displayed on the terminal.} You need a network connection for this.
After that, all you have to do is type ``yes'' in the terminal. Once you installed, there is no further setting up.
On Linux, you can use any editor provided to write a document, such as gedit, emacs, etc. 
Some of them are associated with the software. You can find icons in the tool-bar to compile the file.
No need to worry even if you don't find it. You can always compile 
from the terminal by typing \verb|latex mydoc.tex|. The extension of the file name
must be tex.

If you are using Windows, there is so called Mik{\TeX} as one of the free types.~\cite{miktex} 
After installing the Windows-adapted {\TeX}, you may have to setup 
the editor, but most likely the package includes the editor and compiler after the installation completed.
In order to save and compile, click the related button on the tool-bar.  
It depends on the versions and its setting, but one button could compile it and display the dvi or pdf file without further 
arrangement. While it is compiling, there would appear error messages. You can keep it compiled by 
pressing enter key if you want to see the final result anyway. There is always a button or command 
to abort compiling. For Linux, press control key and `z' at the same time to stop it forcibly.

\section{Basic Settings for a {\TeX} Document}
The document of {\TeX} file starts from \verb|\begin{document}| and ends at \verb|\end{document}|.
The style of the document is set by typing \verb|\documentclass[11pt]{report}| before 
\verb|\begin{document}|\footnote{This is called preamble.}. The options of style can be ``report'', ``article'', ``book'', etc.
The parameter \verb|[11pt]| is the font size. You can also find the following setting if you see the source file:
\begin{verbatim}
\renewcommand{\baselinestretch}{1.5}
\setlength{\topmargin}{.1in}
\setlength{\textheight}{8.5in}
\setlength{\oddsidemargin}{.1in}
\setlength{\textwidth}{6.0in}
\end{verbatim}
The first line determines the line spacing. If the parameter is \{2.0\}, it makes double space.
The rest of the lines set the margin of the document. After this, the several packages are listed, which make 
us utilize more expressions for the document. This web page shows the current available packages.~\cite{pack}
The commands, \verb|\title| and \verb|\author|, are 
shown in the cover page when you command \verb|\maketitle|. This will also give the date after the title.
When you put chapters and sections, you will command with \verb+\chapter{title}+, \verb+\section{title}+, 
\verb+\subsection{title}+, etc. If you want to start a new line forcibly, you have to use double backslash, \verb+\\+.
When you put \verb|\tableofcontents|, the contents are shown before starting the document. 
The table of contents must be updated by compiling the file two times usually.

The style called REV\TeX is the one which you should know if you submit a manuscript to journals of 
American Physical Society.~\cite{revtex}

\section{Reserved Characters for \LaTeX}
The following characters can be used for the text in addition to numbers and alphabetical letters:
\begin{center}
\verb| . : ; , ? ! ( ) [ ] ` ' - / * @|
\end{center}
However, the following characters cannot be used in the text directly since they are reserved 
for {\LaTeX} commands:
\begin{center}
\verb+ # % $ & _ { } ^ ~ \ | < > +
\end{center}
You can find most of the above in the math mode. The \% is used to comment 
in \LaTeX. When you use some of these characters in the sentences, you have to put backslash, 
such as \verb|\# \$ \& \_ \{ \%|. Some of them need \verb|\verb| command, such as 
\verb+\verb|\|+ to express a backslash. You can also use \verb|\backslash| and it shows $\backslash$.
Some of them also have the commands like backslash, such as \verb|\textless| is \textless, \verb|\textgreater| 
is \textgreater, \verb|\textbar| is \textbar, \verb|\textascciicircum| is \textasciicircum, and
\verb|\textasciitilde| is \textasciitilde. Note that \verb+\backslash+ is a math expression, so
you have to use \$ to insert it in the text sentence. This will be explained in the next chapter.

\chapter{Designing Mathematical Expressions}			\label{ch:equation}

{\LaTeX} allows you to express a beautiful mathematical equations.  
Some of journals accept the original {\LaTeX} file. Once you complete it, 
your article can be published without any further editing. Let's get started.

\section{Examples for Displaying Equations}
Especially in physics and math, there are many of the special symbols for equations. For example,
\begin{displaymath}
\vec{B} = \vec{\nabla} \times \vec{A}
\end{displaymath}
Alphabetical symbols are typed directly from the keyboard, but the arrows, cross product, and 
dell operator need some technique. This is the command you will write in the {\LaTeX} file to express above:
\begin{verbatim}
\begin{displaymath}
\vec{B} = \vec{\nabla} \times \vec{A}
\end{displaymath}
\end{verbatim}
Commands start with backslash and the argument will be described inside curly brackets.
\footnote{Some cases use square brackets for the second argument.}
As you can see, \verb|\vec| makes the vector notation. \verb|\nabla| and \verb|\times|
are $\nabla$ and $\times$, respectively.
\verb|\begin{displaymath}| and \verb|\end{displaymath}| indicate the start and end places of 
the mathematical expression. ``Displaymath'' places the equation in the center without numbering.
If you need to label each equation, you have to use ``equation'' as follows: 
\begin{verbatim}
\begin{equation}
\phi' = \phi - \frac{\partial x_{0}}{\partial t}
\end{equation}
\end{verbatim}
This will give the following:
\begin{equation}
\phi' = \phi - \frac{\partial x_{0}}{\partial t}
\end{equation}
As you can see the commands, \verb|\frac{a}{b}| gives $\frac{a}{b}$. You can also use 
\verb|{a \over b}| to express ${a \over b}$.

When you want to display more than one equation continuously with numbering, you have to use ``eqnarray'':
\begin{verbatim}
\begin{eqnarray}
f(x) = \sqrt{\frac{\pi}{2l^2}} \sum^{\infty}_{n=-\infty} e^{in \pi x/l} F_{n} \\
F_{n} = \frac{1}{\sqrt{2 \pi}} \int^{l}_{-l} e^{in \pi x/l} f(x) dx.
\end{eqnarray}
\end{verbatim}
This will display as:
\begin{eqnarray}
f(x) = \sqrt{\frac{\pi}{2l^2}} \sum^{\infty}_{n=-\infty} e^{in \pi x/l} F_{n} \\
F_{n} = \frac{1}{\sqrt{2 \pi}} \int^{l}_{-l} e^{in \pi x/l} f(x) dx.
\end{eqnarray}
Note that there must be \verb|\\| to start the next equation. A superscript 
and subscript are expressed with \verb|^{ }| and \verb|_{ }|, respectively.

While using ``eqnarray'', you can delete the numbers by inserting \verb|\nonumber|:
\begin{verbatim}
\begin{eqnarray}
f(x) = \sqrt{\frac{\pi}{2l^2}} \sum^{\infty}_{n=-\infty} 
	e^{in \pi x/l} F_{n}  \nonumber \\
F_{n} = \frac{1}{\sqrt{2 \pi}} \int^{l}_{-l} e^{in \pi x/l} f(x) dx,
\end{eqnarray}
\end{verbatim}
This will be:
\begin{eqnarray}
f(x) = \sqrt{\frac{\pi}{2l^2}} \sum^{\infty}_{n=-\infty} 
	e^{in \pi x/l} F_{n}  \nonumber \\
F_{n} = \frac{1}{\sqrt{2 \pi}} \int^{l}_{-l} e^{in \pi x/l} f(x) dx,
\end{eqnarray}

You may have multiple equations to list without number labeling. It is not efficient 
to use \verb|\nonumber| for each equation. In this case, use ``eqnarray*''.
\begin{verbatim}
\begin{eqnarray*}
f(x) = \sqrt{\frac{\pi}{2l^2}} \sum^{\infty}_{n=-\infty} e^{in \pi x/l} F_{n} \\
F_{n} = \frac{1}{\sqrt{2 \pi}} \int^{l}_{-l} e^{in \pi x/l} f(x) dx
\end{eqnarray*}
\end{verbatim}
Then it shows:
\begin{eqnarray*}
f(x) = \sqrt{\frac{\pi}{2l^2}} \sum^{\infty}_{n=-\infty} e^{in \pi x/l} F_{n} \\
F_{n} = \frac{1}{\sqrt{2 \pi}} \int^{l}_{-l} e^{in \pi x/l} f(x) dx
\end{eqnarray*}

If you want to include equations or symbols in the text sentences, you will	
use dollar signs or \verb+\(...\)+. For example,
\begin{center}
The function is expressed as \verb|$j_{0}(\rho)=\frac{\sin \rho}{\rho}$|.
 \end{center}
That will give:
\begin{center}
The function is expressed as $j_{0}(\rho)=\frac{\sin \rho}{\rho}$.
 \end{center}
Also, $\sqrt{2}=1.41421356...$, and other special characters like $\Box$, $\Diamond$, $\mho$, $\clubsuit$, 
and $\oplus$ have to use \verb|$$|  in the sentences. Here is a summary and 
extra equivalent commands:
\begin{table}[h]
  \begin{tabular*}{17cm}{@{\extracolsep{\fill}} l|l|l|l}
Purpose			&Command 1			&Command 2						&Command 3 \\ 
\hline \hline
In text sentences		&\verb|$...$|		 	&\verb+\begin{math}+\ldots \verb+\end{math}+ 	&\verb+\(...\)+ \\
\hline
Display 			&\verb|$$...$$|	 	&\verb+\begin{displaymath}+		 	&\verb+\[...\]+ \\
without number		&				&\ldots \verb+\end{displaymath}+			&		  \\
\hline
Single equation		& 				&\verb+\begin{equation}+ 				&		\\
with numbers 			&				&...\verb+\end{equation}+ 				&		 \\
\hline
Multiple equations		& 				&\verb+\begin{eqnarray}+				&		 \\
with numbers			&				&...\verb+\end{eqnarray}+ 				&		 \\
\hline
Multiple equations		&	 			&\verb+\begin{eqnarray*}+				&		 \\
without numbers		&				&...\verb+\end{eqnarray*}+ 			& 		\\
\hline \hline
  \end{tabular*}
\end{table}

The equations in array may have different lengths, such as following:
\begin{eqnarray}
f(x) = \sqrt{\frac{\pi}{2l^2}} \sum^{\infty}_{n=-\infty} e^{in \pi x/l} F_{n} \\
k=\frac{n \pi}{l}
\end{eqnarray}
The equations are aligned to the right. In order to realign them with $=$ signs, use \verb|&&| 
to `clip' them as follows:
\begin{verbatim}
\begin{eqnarray}
f(x) &=& \sqrt{\frac{\pi}{2l^2}} \sum^{\infty}_{n=-\infty} e^{in \pi x/l} F_{n} \\
k    &=& \frac{n \pi}{l}
\end{eqnarray}
\end{verbatim}
Then, we can have a better look for the array of equations:
\begin{eqnarray}
f(x) &=& \sqrt{\frac{\pi}{2l^2}} \sum^{\infty}_{n=-\infty} e^{in \pi x/l} F_{n} \\
k    &=& \frac{n \pi}{l}
\end{eqnarray}
The above technique can be used with any part of the equation besides $=$ signs.
You can also arrange two equations in one line if they are short enough to them together: 
\begin{equation}
\tan 2 \gamma = -b \cos 2 \gamma , \tan 2 \gamma = -\frac{2b}{\Delta}
\end{equation}
It looks alright; however, the space between two equations looks too close each other. In the 
math mode, simple spaces made by the space key will not be counted.
One of the methods is a combination of backslash and semicolon that makes 
a certain space between equations:
\begin{verbatim}
\begin{equation}
\tan 2 \gamma = -b \cos 2 \gamma ,  \;\;\;\;\; 
		\tan 2 \gamma = -\frac{2b}{\Delta}.
\end{equation}
\end{verbatim}
You can adjust the range of space by using multiple \verb|\;|, and this gives:
\begin{equation}
\tan 2 \gamma = -b \cos 2 \gamma ,  \;\;\;\;\; 
		\tan 2 \gamma = -\frac{2b}{\Delta}.
\end{equation}
There are several ways to create sizes of space. Here is a table:
\begin{table}[h]
  \begin{tabular*}{15cm}{@{\extracolsep{\fill}} llc}
	Commands&		Size of space				&Note \\ \hline \hline
     	\verb*|\ |&		Single-byte		 	& in both math and text modes \\
    	 \verb|\quad|&	Double-byte		 	& in both math and text modes \\
    	\verb|\qquad|&	Twice quad			& in both math and text modes \\
    	 \verb|\,|&		One sixth of quad	 	& in both math and text modes \\
    	 \verb|\>|&		Two ninths of quad 		& only in math mode \\
    	 \verb|\;|&		Five eighteenths of quad	& only in math mode \\
    	 \verb|\!|&		One sixth of quad		& only in math mode  \\ \hline \hline
  \end{tabular*}
\end{table}

It is necessary to adjust the size of parentheses and brackets sometimes depending on the equations. Let us consider 
the following:
\begin{equation}
(\frac{\sin\theta}{\tan\theta}-1)=\gamma
\end{equation}
The simple parentheses don't fit well as shown. In order to adjust them automatically, 
you should use \verb|\left(| and \verb|\right)|. For instance, 
\begin{verbatim}
\begin{equation}
\left(\frac{\sin\theta}{\tan\theta}-1\right)=\gamma
\end{equation}
\end{verbatim}
Then it will show:
\begin{equation}
\left(\frac{\sin\theta}{\tan\theta}-1\right)=\gamma
\end{equation}
This command can be used for the other brackets, such as \verb+[+, \verb+ {+, and \verb+|+.
If you want to specify the size of bracket, you can also use the following commands:
\verb|\big(|, \verb|\Big(|, and \verb|\bigg(|, and they look like
$\big($, $\Big($, and $\bigg($. It also depends on the type of {\LaTeX}, its version, 
and packages, so check out each manual when you have any errors.

Expressing chemical reactions is an application of subscripts, but expressions of nuclear reactions need 
superscripts and subscripts opposite side.
Here are a couple of examples:
\begin{eqnarray*}
  2\mathrm{H}_2 \mathrm{O} &\longrightarrow& 2\mathrm{H}_2 + \mathrm{O}_2 \\
  {}^{241}_{95}\mathrm{Am} &\longrightarrow& {}^{237}_{93}\mathrm{Np} 
	+ {}^{4}_{2}\mathrm{He}
\end{eqnarray*}
The following is the way you would command:
\begin{verbatim}
\begin{eqnarray*}
  2\mathrm{H}_2 \mathrm{O} &\longrightarrow& 2\mathrm{H}_2 + \mathrm{O}_2 \\
  {}^{241}_{95}\mathrm{Am} &\longrightarrow& {}^{237}_{93}\mathrm{Np} 
	+ {}^{4}_{2}\mathrm{He}
\end{eqnarray*}
\end{verbatim}

The square-root command takes two arguments. One of them is usually neglected 
when it is a normal square root. In {\LaTeX}, they are expressed as
\begin{verbatim}
\begin{displaymath}
  \sqrt{x^2+y^2+z^2}, \qquad \qquad \sqrt[a]{3b^4}
\end{displaymath}
\end{verbatim}
As you see, the index of the root is described by square brackets and the radicand 
is placed in curly brackets. The above will be:
\begin{displaymath}
  \sqrt{x^2+y^2+z^2}, \qquad \qquad \sqrt[a]{3b^4}
\end{displaymath}

\section{Arrays and Matrices}
The following is one of the simple arrays:
\[ \begin{array}{clcr}
	a+b+c & uv      &  x-y   &   27     \\
	a+b    & u+v    &  z       &   112   \\
            a        & 3u-vw & xyz^2 &  2,338
      \end{array}  \]
The {\LaTeX} expression is:
\footnote{You can use the ``displaymath'' mode instead of $\backslash$[...$\backslash$].}
\begin{verbatim}
\[ \begin{array}{clcr}
	a+b+c & uv      &  x-y   &   27     \\
	a+b    & u+v    &  z       &   112   \\
 a      & 3u-vw &  xyz^2  &  2,338
   \end{array}  \]
\end{verbatim}
An array or matrix starts with \verb|\begin{array}| and ends at 
\verb|\end{array}|. This has arguments shown as \verb|{clcr}|.
This is an alignment information for each column.  
\verb|l|, \verb|c|, and \verb|r| indicates ``left'', ``center'', and 
``right'', respectively. As you can see, each column is aligned as 
designated. Also note that ``\verb|&|'' and ``\verb|\\|'' separate 
each column and raw, respectively.

The following is a little complicated, but a good example to 
understand how arrays work this way:
\[ x- \begin{array}{c}
	a_{1}  \\   \vdots    \\    a_{n}
	\end{array}
          -      \begin{array}[t]{cl}
	  u-v  &  13  \\
	  u+w &  \begin{array}[b]{r}
			12  \\  -982
		    \end{array}
	\end{array}	\]		
The above is expressed in {\LaTeX} as:
\begin{verbatim}
\[ x- \begin{array}{c}
	a_{1}  \\   \vdots    \\    a_{n}
	\end{array}
          -      \begin{array}[t]{cl}
	  u-v  &  13  \\
	  u+w &  \begin{array}[b]{r}
			12  \\  -982
		    \end{array}
	\end{array}	\]	
\end{verbatim}
There are three arrays in this expression. The first one consists of $a_1$, $\vdots$, and 
$a_2$ as a one-column array. The other one is basically $2\times 2$ array and it also contains
an array inside.

The next two arrays are the application from what you have learned so far.
\[   \left(  \begin{array}{c}
		\left|  \begin{array}{cc}
			x_{11} & x_{12}   \\
			x_{21} & x_{22}
			\end{array}
		\right|   \\
		y \\ z
	         \end{array}           \right)    \]
This is coded as follows:
\begin{verbatim}
\[   \left(  \begin{array}{c}
		\left|  \begin{array}{cc}
			x_{11} & x_{12}   \\
			x_{21} & x_{22}
			\end{array}
		\right|   \\
		y \\ z
   \end{array}           \right)    \]
\end{verbatim}
The next example is:
\[  x = \left\{  \begin{array}{ll}
		y   &   \mbox{if $y>0$}  \\
		z+y  & \mbox{otherwise}
		\end{array}
	\right.   \]
The above expression will be written as follows:
\begin{verbatim}
\[  x = \left\{  \begin{array}{ll}
		y   &   \mbox{if $y>0$}  \\
		z+y  & \mbox{otherwise}
		\end{array}
	\right.   \]
\end{verbatim}
The command, \verb|\mbox|, makes us use text characters in the equation mode.
Also note that the dot next to \verb|\right| command, which is necessary not to have right side of bracket.

\section{Accents and Constructs for Math}
For an expression of boolean algebra, you can have the following as an example:
\begin{equation}
  \overline{\overline{A}+B}=\overline{\overline{A}} \cdot \overline{B} 
\end{equation}
This is the {\LaTeX} description of the above:
\begin{verbatim}
\begin{equation}
  \overline{\overline{A}+B}=\overline{\overline{A}} \cdot \overline{B} 
\end{equation}
\end{verbatim}
The command, \verb|\underline|, can also be used in the text mode. For example, 
\verb|\underline{important}| gives \underline{important}.
The other example is:
\begin{equation}
\underbrace{a + \overbrace{ b + \cdots + y}^{35} +z}_{52}
\end{equation}
The {\LaTeX}  is:
\begin{verbatim}
\begin{equation}
\underbrace{a + \overbrace{ b + \cdots + y}^{35} +z}_{52}
\end{equation}
\end{verbatim}
You can also express $\widehat{a+b}$ with \verb~\widehat{a+b}~, etc. 
The command, \verb~a \stackrel{f}{\longrightarrow} b~, will give you:
 $a \stackrel{f}{\longrightarrow} b$. \verb|\stackrel| takes two arguments. The first one is placed 
above and the second is placed below.
Here is the summary:
\begin{table}[htb]
\begin{center}
  \begin{tabular*}{10cm}{@{\extracolsep{\fill}} ll}
	Commands							&Display \\ \hline \hline
	\verb~\widehat{a+b}~ 			& $\widehat{a+b}$ \\
	\verb~\widetilde{a+b}~ 			& $\widetilde{a+b}$ \\
	\verb~\overleftarrow{a+b}~ 		& $\overleftarrow{a+b}$ \\
	\verb~\overrightarrow{a+b}~ 		& $\overrightarrow{a+b}$ \\ 
	\verb~\overbrace{a+b}~ 			& $\overbrace{a+b}$ \\ 
	\verb~\underbrace{a+b}~ 			& $\underbrace{a+b}$ \\ 
	\verb~\overline{a+b}~ 			& $\overline{a+b}$ \\ 
	\verb~\underline{a+b}~ 			& $\underline{a+b}$ \\ 
\hline \hline
  \end{tabular*}
\end{center}
\end{table}

\section{Styles in Math Mode}
In the \underline{math mode}, {\LaTeX} has specific commands for various fonts and styles, which are shown in the next table.
The following five fonts can be used for upper- and lower-case English letters, numbers, and upper-case Greek letters.
\begin{table}[htb]
  \begin{tabular*}{15cm}{@{\extracolsep{\fill}} lcr}
	Fonts&		Commands					&Display \\ \hline \hline
     	Italic&		\verb|\mathit{ABCabc123\Gamma\Pi}| 	& $\mathit{ABCabc123\Gamma\Pi}$ \\
    	 Roman& 	\verb|\mathrm{ABCabc123\Gamma\Pi}| 	& $\mathrm{ABCabc123\Gamma\Pi}$ \\
    	 Bold face&	\verb|\mathbf{ABCabc123\Gamma\Pi}|	& $\mathbf{ABCabc123\Gamma\Pi}$ \\
    	 Sans Serif&	\verb|\mathsf{ABCabc123\Gamma\Pi}| 	& $\mathsf{ABCabc123\Gamma\Pi}$ \\
    	Typewriter&	\verb|\mathtt{ABCabc123\Gamma\Pi}| 	& $\mathtt{ABCabc123\Gamma\Pi}$ \\ \hline \hline
  \end{tabular*}
\end{table}
For example, in physics, you may find the following expression:
\begin{displaymath}
{\cal{L}} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{c}j_{\mu}A^{\mu}
\end{displaymath}
This is the Lagrangian density of electromagnetic fields and the symbol of Lagrangian is expressed in a calligraphic 
style. The {\LaTeX} description is:
\begin{verbatim}
\begin{displaymath}
{\cal{L}} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{c}j_{\mu}A^{\mu}
\end{displaymath}
\end{verbatim}
You have to use \verb|\cal| with another set of curly brackets such as \verb|{\cal{L}}|; 
otherwise, other letters will be converted due to \verb|\cal|.\footnote{$\backslash$mathcal
command does not need this treatment.}  Also 
we should note that \verb|\cal| command is only for upper case letters of alphabet. If you 
use it with a lower case letter, \verb|{\cal{a}}|, you will get a different character like ${\cal{a}}$. Other types of 
fonts are shown in the next table.
\begin{table}[htb]
  \begin{tabular*}{15cm}{@{\extracolsep{\fill}} lcr}
	Fonts&		Commands				&Display \\ \hline \hline
     	Calligraphic &	\verb|\mathcal{ABCDEFG}|	 	& $\mathcal{ABCDEFG}$ \\
    	Mathbb& 	\verb|\mathbb{ABCDEFG}|	 	& $\mathbb{ABCDEFG}$ \\   
    	Mathfrak&	\verb|\mathfrak{ABCabc123}|	& $\mathfrak{ABCabc123}$ \\  \hline \hline
  \end{tabular*}
\end{table}
Calligraphic and Mathbb take only uppercase English letters. Mathfrak takes both upper and lower cases of 
English letters and numbers, but not Greek letters. These are used only in the math mode.

You can also change the font sizes in the math mode. Here are the commands:
\begin{table}[htb]
  \begin{tabular*}{15cm}{@{\extracolsep{\fill}} lc}
	Commands							&Display \\ \hline \hline
	\verb+{\scriptscriptstyle \oint \vec{B}\cdot d\vec{A}=0}+ 	& ${\scriptscriptstyle \oint \vec{B}\cdot d\vec{A}=0}$ \\
	\verb+{\scriptstyle \oint \vec{B}\cdot d\vec{A}=0}+ 	& ${\scriptstyle \oint \vec{B}\cdot d\vec{A}=0}$ \\
	\verb+{\textstyle \oint \vec{B}\cdot d\vec{A}=0}+ 	& ${\textstyle \oint \vec{B}\cdot d\vec{A}=0}$ \\
	\verb+{\displaystyle \oint \vec{B}\cdot d\vec{A}=0}+ 	& ${\displaystyle \oint \vec{B}\cdot d\vec{A}=0}$ \\ \hline \hline
  \end{tabular*}
\end{table}

\section{Referring to Equations}
It is not a smart idea to refer to many of equations by looking at the compiled dvi (or pdf) file 
to check out the number each time. You may have to relabel all of the numbers when you 
add or delete some of the equations. The label command can register each equation to refer. 
Let us see an example:
\begin{verbatim}
\begin{equation}
i \hbar \frac{\partial}{\partial t}\Psi = \left[ -\frac{\hbar^2}{2m}\nabla^2
         +V \right]\Psi                   \label{eq:schr}
\end{equation}
\end{verbatim}
The above will give you the following:
\begin{equation}
i \hbar \frac{\partial}{\partial t}\Psi = \left[ -\frac{\hbar^2}{2m}\nabla^2
         +V \right]\Psi                  \label{eq:schr}
\end{equation}
You can find \verb+\label{  }+ in the equation. The name can be anything. The 
reason why ``eq:'' is put in the label usually because it is convenient  
to tell that it is referring to an equation. If you want to refer to the equation in the sentences, you will 
put \verb+~(\ref{eq:schr})+ in an appropriate place. This is an example: 
\begin{quote}
The Schr\"{o}dinger equation \verb+~(\ref{eq:schr})+ describes mechanics of quantum regions.
\end{quote}
Then, you will see as
\begin{quote}
The Schr\"{o}dinger equation~(\ref{eq:schr}) describes mechanics of quantum regions.
\end{quote}
You can also refer to tables, figures, and other locations. 
The table of cross-reference commands is shown in Table~\ref{tb:cref}.

\chapter{Formatting Texts}					\label{ch:text}
\section{Special Characters and Symbols}
Languages other than English have a variety of accents 
and special symbols to express their particular pronunciations.

\begin{itemize}
\item Accents and special pronunciations

Spanish, French, German and other languages have accents and particular 
pronunciations. Espa\~{n}ol and Amp\`{e}re are the examples.
These are expressed in {\LaTeX} as \verb|Espa\~{n}ol| and \verb|Amp\`{e}re|.
You have to type backslash and a symbol right before the letter to modify. The letter 
has to be in the curly brackets. Other accents are shown in the following table:

\begin{table}[h]
\begin{center}
  \begin{tabular}{cc}
			Commands		&	Display \\ \hline \hline
     		\verb|\'{E}l est\'{a} aqu\'{i}| 	& 	\'{E}l est\'{a} aqu\'{i} \\
    	 	\verb| Schr\"{o}dinger| 	& 	 Schr\"{o}dinger \\
		\verb|M\oller|  &			M\o ller \\	
    		\verb|Gau\ss| 	& 			Gau\ss \\ \hline
		\verb|\oe|  &				\oe   \\
		\verb|\aa|  &				\aa   \\
		\verb|\AA|  &				\AA   \\
		\verb|\ae|  &				\ae   \\
		\verb|\={o}|  &			\={o}   \\
		\verb|\c{o}|  &			\c{o}   \\
		\verb|\t{oo}|  &			\t{oo}   \\ \hline \hline
  \end{tabular}
  \caption{Accents and Pronunciations}
\end{center}
\end{table}

\item Other special symbols

The other symbols and special punctuations are shown in the next table: 
\begin{table}[h]
\begin{center}
  \begin{tabular}{cc}
		Commands&				Display \\ \hline \hline
		\verb|\dag|  &				\dag   \\
		\verb|\ddag|  &			\ddag   \\
		\verb|\S|  &				\S   \\
		\verb|\copyright|  &			\copyright   \\
		\verb|\P|  &				\P   \\
		\verb|\pounds|  &			\pounds   \\
		\verb|!`|  &				!`   \\
		\verb|?`|  &				?`   \\ \hline \hline
  \end{tabular}
  \caption{Special Symbols}
\end{center}
\end{table}

\end{itemize}

\section{Fonts, Sizes, etc.}
In the text mode, the fonts and the sizes are changed as follows:
\begin{table}[h]
\begin{center}
  \begin{tabular}{cc|cc}
			Commands			&Display 	            &Commands			&Display\\ \hline \hline
		\verb|\tiny|  &				{\tiny tiny}   		&\verb|\rm|			&{\rm{roman}} \\
		\verb|\scriptsize|  &			{\scriptsize script size}&\verb|\tt|			&{\tt{typewriter}}\\
		\verb|\footnotesize|  &		{\footnotesize footer}	&\verb|\sf|			&{\sf{sans serif}}\\
		\verb|\small|  &			{\small small}  		&\verb|\sc|			&{\sc{Small Caps}}   \\
		\verb|\nomalsize|  &			{\normalsize normal}	&\verb|\bf|			&{\bf{bold face}}   \\
		\verb|\large|  &			{\large large}		&\verb|\it|			&{\it{italic}}\\
		\verb|\Large|  &			{\Large Large}		&\verb|\sl|			&{\sl{slant}} \\ 
		\verb|\LARGE| &			{\LARGE LARGE}	&\verb|\em|			&{\em emphasized}	\\
		\verb|\huge| &			{\huge huge}		&				&	        \\ 
		\verb|\Huge| &			{\Huge Huge}		&				&	       \\  \hline \hline
  \end{tabular}
  \caption{Fonts and sizes}
\end{center}
\end{table}


\section{Tables}
Here is a typical table for a science journal and its {\LaTeX} commands are described below:
\begin{table}[h]
\begin{center}
\begin{tabular}{ccccc}\hline
$m$ (kg) & $x$ and $l$(m) & $T_S$ (s)    & $T_P$ (s) & \% difference (\%) \\ 
\hline \hline
0.200         & 0.293          & 1.091       & 1.088        & 0.303   \\ 
0.300         & 0.438          & 1.329       & 1.329        & 0.015   \\ 
0.400         & 0.583          & 1.525       & 1.533        & 0.576   \\ \hline
\end{tabular}
\caption{$m$ is the hanging mass.  $x$ is the displacement of the spring with 
a certain hanging mass...}
\end{center}
\end{table}
\begin{verbatim}
\begin{table}[h]
\begin{center}
\begin{tabular}{ccccc}\hline
$m$ (kg) & $x$ and $l$(m) & $T_S$ (s)    & $T_P$ (s) & \% difference (\%)  \\ 
\hline \hline
0.200         & 0.293          & 1.091       & 1.088        & 0.303   \\ 
0.300         & 0.438          & 1.329       & 1.329        & 0.015   \\ 
0.400         & 0.583          & 1.525       & 1.533        & 0.576   \\ \hline
\end{tabular}
\caption{$m$ is the hanging mass.  $x$ is the displacement of the spring with 
a certain hanging mass...}
\end{center}
\end{table}
\end{verbatim}
The usage of \& and \verb+\\+ are the same as making arrays. \& is to separate the columns and 
\verb+\\+ is to start the next raw. In order to center the table, you have to use the center command as shown above.
It is important to place it right after the table command. The \verb|\hline| command is for drawing a horizontal line.
For the argument of the tabular command, \verb|{ccccc}|, this is also the same as the array commands.
You need to put each information of the alignment for each column.\footnote{l, c, and r represent left, center, and right, respectively.} For the table command, there is an argument, 
\verb+[h]+. There are four different parameters: These specify the preferable location of the table to be displayed. 
They are \verb|h|, \verb|t|, \verb|b|, and \verb|p|, which indicates ``here'', ``top'', ``bottom'', and ``page'', respectively.
If you use \verb|p|, the tables will be located in particular pages only for tables. If you want to prioritize 
your request, you will put \verb|!| next to the parameter, such as \verb|[h!]|. 
\footnote{The {\LaTeX} compiler tries to listen to your request, but it doesn't necessarily mean that 
it arranges exactly as you expect.} Also you can put every parameter if you don't have any priority. 
The caption can be included with the caption command as you see in the example.

Two more basic techniques will be explained here. 
Some case needs to draw vertical lines in a table. This doesn't require any command. 
Simply put \underline{vertical bars} in the tabular 
option. In the above example, if you want to have a vertical line between the second and the third columns, 
you can make it as, \verb+{cc|ccc}+. You can see other examples:
\begin{table}[h!]
  \begin{center}
    \begin{tabular}{c}

      % 1
      \begin{minipage}{0.25\hsize}
        \begin{center}
          \caption{\{$|$cc\}}
          \begin{tabular}{|cc}
            \hline
            A & B  \\ \hline \hline
            1 & 0  \\
            0 & 1  \\
            \hline
          \end{tabular}
        \end{center}
      \end{minipage}

      % 2
      \begin{minipage}{0.25\hsize}
        \begin{center}
          \caption{\{c$|$c\}}
          \begin{tabular}{c|c}
            \hline
            A & B  \\ \hline \hline
            0 & 0  \\
            0 & 1  \\

            \hline
          \end{tabular}
        \end{center}
      \end{minipage}

      % 3
      \begin{minipage}{0.25\hsize}
        \begin{center}
          \caption{\{$|$cc$|$\}}
          \begin{tabular}{|cc|}
            \hline
            A & B  \\ \hline \hline
            1 & 0  \\
            0 & 0  \\

            \hline
          \end{tabular}
        \end{center}
      \end{minipage}

      % 4
      \begin{minipage}{0.25\hsize}
        \begin{center}
          \caption{\{$|$c$|$c$|$\}}
          \begin{tabular}{|c|c|}
            \hline
            A & B  \\ \hline \hline
            0 & 1  \\
            0 & 0  \\

            \hline
          \end{tabular}
        \end{center}
      \end{minipage}
    \end{tabular}
  \end{center}
\end{table}

Some item of a table may take multiple columns and raws. The following is an example:
\begin{table}[h]
  \begin{center}
   \begin{tabular}{c|c|l}
   \hline
   \multicolumn{3}{c}{Title}\\
   \hline\hline
   \multicolumn{2}{c|}{\bf Items 1 and 2} & {\bf Item 3} \\
   \hline
   A & B & Category 1 \\
   \cline{1-2}
   I & II & \\
   \cline{1-2}
   \hline
   $\alpha$ & $\beta$ & Category 2 \\
   \hline
   \end{tabular}
  \end{center}
  \caption{Multiple columns and raws}
\end{table}

\begin{verbatim}
\begin{table}[h]
  \begin{center}
   \begin{tabular}{c|c|l}
   \hline
   \multicolumn{3}{c}{Title}\\
   \hline\hline
   \multicolumn{2}{c|}{\bf Items 1 and 2} & {\bf Item 3} \\
   \hline
   A & B & Category 1 \\
   \cline{1-2}
   I & II & \\
   \cline{1-2}
   \hline
   $\alpha$ & $\beta$ & Category 2 \\
   \hline
   \end{tabular}
  \end{center}
  \caption{Multiple columns and raws}
\end{table}
\end{verbatim}
The title cell is using 3 columns. The command is \verb|\multicolumn{3}{c}{Title}|.
For multiple raws for one item, you skip the item that you want to exclude and draw horizontal line from 
the certain column to the specified column by using \verb|\cline{ }|. In the above example, it is designated as \{1-2\}.
This means that the horizontal line is drawn from the first to second columns.

\section{Quotations and Theorems}
Here is an example with the quote or quotation command:
\begin{verbatim}
\begin{quote}
It takes decades to realize educational effect. Country's prosperity comes from 
a long term efforts on education.
\end{quote}
\end{verbatim}
This will be displayed from the above description:
\begin{quote}
It takes decades to realize educational effect. Country's prosperity comes from 
a long term efforts on education.
\end{quote}

You can use \verb+\begin{center}...\end{center}+ to express a statement in 
a different paragraph in the center although this is not only for quotation.

In order to express a theorem-like statement, you will use \verb+\newtheorem+ to define the name of the theorem:  
\begin{verbatim}
\newtheorem{guess}{Conjecture}
\begin{guess}
All conjectures are interesting, but some conjectures 
are more interesting than others.
\end{guess}
\end{verbatim}
The first argument in ``newtheorem'' is the name of the environment, and
the second one is the statement to label the entire theorem. The above will show you following:
\newtheorem{guess}{Conjecture}
\begin{guess}
All conjectures are interesting, but some conjectures 
are more interesting than others.
\end{guess}

When you quote a word with double-quotation marks, you may use ( " ) on the keyboard.
The following is the example of using the simple double quotations:
\begin{center}
"quoted"
\end{center}
It looks alright, but you can notice that this is not a correct pair. In {\LaTeX}, you have to 
use twice ( ` ) and twice ( ' ) to open and close the quotation. This ( ` ) is located under the escape key.
This ( ' ) is the normal apostrophe. Then, you have:
\begin{center}
``quoted''
\end{center}
Now, it looks right.

\section{Marginal Notes and Footnotes}
A marginal note is displayed with the command, \verb+\marginpar{ }+. Your comment 
will be inside the curly brackets. For example, \verb|\marginpar{This is a marginal note.}|.
\marginpar{This is a marginal note.} This will be appeared in the margin.

You can also comment in the footer by using \verb|\footnote{ }|. This can be used right after 
the sentence you want to refer to. The command is, foe example,  \verb|\footnote{The footnote is appeared here!}|.
\footnote{The footnote is appeared here!} This is labeled as a reference and
you can see it at the bottom of the page.

\section{Numbering and Bulleting}
Listing some of items in a paragraph is required to have commands for bullets or numbers.
For bulleting, you will set the range by using the itemize command.
For example,
\begin{verbatim}
\begin{itemize}
\item This is what I want to say first.
\item This is what I want to say second.
\end{itemize}
\end{verbatim}
The command, \verb|\item|, makes a bullet in this range. 
\begin{itemize}
\item This is what I want to say first.
\item This is what I want to say second.
\end{itemize}
If you want to use numbers, you will use ``enumerate'' instead of ``itemize.''
The example is following:
\begin{verbatim}
\begin{enumerate}
\item The first item
\item The second item
\end{enumerate}
\end{verbatim}
This will display:
\begin{enumerate}
\item The first item
\item The second item
\end{enumerate}
You can also use your own label by using the description command:
\begin{verbatim}
\begin{description}
\item [$\star$] This is a star item.
\item [Label]  It becomes bold.
\end{description}
\end{verbatim}
This will give:
\begin{description}
\item [$\star$] This is a star item.
\item [Label]  It becomes bold.
\end{description}
The description command is a little different from the numbering and bulleting.
The labeled items are not tabbed.

\chapter{Inserting Figures}					\label{ch:figure}
Some of the commands shown here may depend on operating systems, the version of {\LaTeX}, and 
the packages used. It is recommended that you 
refer to the correspondent manual. For instance, some picture formats cannot be recognized by 
the {\LaTeX}. On a Linux operating system, people usually 
make use of an encapsulated PostScript file for figures.
The extension is denoted as eps.\footnote{The other formats can be recognized, but it depends on the operating system and {\LaTeX}.}
In order to use the figure commands, you have to use 
\verb+\usepackage{graphicx}+; and then, put the picture files into the same folder.  
The following is a basic set of commands to allocate a picture file in the document:
\begin{verbatim}
\begin{figure}[htbp]
  \begin{center}
   \includegraphics[width=4cm]{mypicture.eps}
   \caption{Picture 1}
   \label{fig:pic1}
  \end{center}
\end{figure}
\end{verbatim}
 \marginpar{The command, $\backslash$label\{ \} is for the cross reference of the figure.}
It will start with and end at the figure command. The command, \verb|\includegraphics|, 
takes the arguments to determine the size and to indicate the picture file to display. There are more options besides \verb|width|:
The command, \verb|height|, adjusts the height of the figure; \verb|scale| changes the ratios; \verb|angle| rotates 
the figure; and \verb|clip| trims the original picture file when you make it \verb|clip=true|. You can put multiple 
parameters using commas. If you want to center the 
picture, you will use the center command. The same picture file with two different sizes are posted as follows:

\begin{figure}[h!]
  \begin{center}
        \includegraphics[width=16cm]{constellations.jpg}
          \hspace{1.6cm} 16cm-width
  \end{center}
\end{figure}
\begin{figure}[h!]
  \begin{center}
          \includegraphics[width=8cm]{constellations.jpg}
          \hspace{0cm} 8cm-width
 
    \caption{Different sizes with the same picture}
  \end{center}
\end{figure}


\chapter{Making Bibliography}					\label{ch:bib}
Bib{\TeX} is a convenient command to manage the list of references in a separate file.  
There is another command, ``thebibliography'', which makes us include the references in the same {\LaTeX} file. 
Similarly to other commands, it starts with and end at the ``thebibliography'' command. For instance,
\begin{verbatim}
\begin{thebibliography}{99}
\bibitem{Razavy} Razavy, Mohsen. 2005. 
          \textit{Classical and Quantum Dissipative Systems}. Imperial College Press
\end{thebibliography}
\end{verbatim}
After the \verb|thebibliography| command, there is a parameter, \verb+{99}+. This indicates that the maximum 
reference number would have up to 2 figures. The \verb+\bibitem{ }+ command labels the name for citation. 
The \verb+\bibitem+ takes two arguments, such as \verb+\bibitem[remark]{key}+. The ``remark'' is the name shown as the reference. 
You can remove the remark as in the example above; then, the number will be appeared as the reference instead. 
For the cress-reference, you will use \verb+~\cite{Razavy}+ at the proper place in the document 
when you need to cite the reference.

The following shows a preamble for the format of American Physical Society. This can be copied
and pasted into your {\TeX} file or the homepage of APS may provide an example source file.
\begin{verbatim}
\begin{thebibliography}{99}

%%%%%%    preamble    %%%%%%
\expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi
\expandafter\ifx\csname bibnamefont\endcsname\relax
  \def\bibnamefont#1{#1}\fi
\expandafter\ifx\csname bibfnamefont\endcsname\relax
  \def\bibfnamefont#1{#1}\fi
\expandafter\ifx\csname citenamefont\endcsname\relax
  \def\citenamefont#1{#1}\fi
\expandafter\ifx\csname url\endcsname\relax
  \def\url#1{\texttt{#1}}\fi
\expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi
\providecommand{\bibinfo}[2]{#2}
\providecommand{\eprint}[2][]{\url{#2}}

%%%%%%    start bibliography    %%%%%%
\bibitem[{\citenamefont{Rut11}}]{Rut11}
\bibinfo{author}{\bibfnamefont{E.}~\bibnamefont{Rutherford}},
  \bibinfo{journal}{Phil. Mag.} \textbf{\bibinfo{volume}{{\bf{21}}}},
  \bibinfo{pages}{669} (\bibinfo{year}{1911}).
\end{thebibliography}
\end{verbatim}

The source file of this document is also provided. Please refer to how \verb|\thebibliography| is 
used in this {\LaTeX} file.

\chapter{Command Reference}					\label{ch:ref}
This chapter provides useful symbols, letters, and commands as a reference.
In order to express some symbols, it requires packages, \verb|\usepackage{amsmath}| and \verb|\usepackage{amssymb}|. 
The symbols introduced here are not all of the available ones. You can find more of them in some web pages
and various books. A couple of documents~\cite{UCB,UK} are useful to refer to more symbols.

\begin{table}[h!]
  \begin{tabular*}{17cm}{@{\extracolsep{\fill}} cccc}
{\bf Greek Letters} 		&				&					&\\  
(lower cases)			&				&					&\\ \hline
$\alpha$  \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|	&				 	&$\varkappa$ \verb|\varkappa| \\ \hline
  \end{tabular*}
\end{table}

\begin{table}[h!]
  \begin{tabular*}{17cm}{@{\extracolsep{\fill}} cccc}
{\bf Greek Letters} 		&				&					&\\  
(upper cases)			&				&					&\\ \hline
$\Gamma$  \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|	 	&$\digamma$ \verb|\digamma| \\ \hline
  \end{tabular*}
\end{table}

\begin{table}[h!]
  \begin{tabular*}{17cm}{@{\extracolsep{\fill}} cccc}
{\bf Operation Symbols}	&				&					&\\  \hline
$\pm$  \verb|\pm|		&$\mp$  \verb|\mp|		& $\times$  \verb|\times|	 	&$\div$  \verb|\div| \\
$\ast$  \verb|\ast|		&$\star$  \verb|\star|		& $\circ$  \verb|\circ| 			&$\bullet$  \verb|\bullet| \\
$\cdot$  \verb|\cdot|	 	&$\cap$  \verb|\cap|		& $\cup$  \verb|\cup|		 	&$\vee$  \verb|\vee| \\
$\wedge$  \verb|\wedge| &$\wr$ \verb|\wr| &$\bigtriangleup$  \verb|\bigtriangleup| & $\bigtriangledown$  \verb|\bigtriangledown| \\		
$\lhd$  \verb|\lhd|		&$\rhd$  \verb|\rhd|		& $\unlhd$  \verb|\unlhd|	 	&$\unrhd$  \verb|\unrhd| \\
$\oplus$  \verb|\oplus|	&$\ominus$  \verb|\ominus|	& $\otimes$  \verb|\otimes| 		&$\oslash$  \verb|\oslash| \\
$\odot$  \verb|\odot|		&$\bigcirc$  \verb|\bigcirc|	& $\dagger$  \verb|\dagger| 		&$\ddagger$  \verb|\ddagger| \\
$\uplus$  \verb|\uplus|	&$\sqcap$  \verb|\sqcap|	& $\sqcup$  \verb|\sqcup| 		&$\triangleleft$  \verb|\triangleleft| \\
$\triangleright$  \verb|\triangleright|	&$\diamond$  \verb|\diamond|&$\Box$ \verb| \Box|		&$\amalg$ \verb|\amalg|   \\
\hline
  \end{tabular*}
\end{table}

\begin{table}[h!]
  \begin{tabular*}{17cm}{@{\extracolsep{\fill}} cccc}
{\bf Relation Symbols}	&				&					&\\  \hline
$\propto$  \verb|\propto|	&$\ll$  \verb|\ll|		& $\gg$  \verb|\gg|	 		&$\subset$  \verb|\subset| \\
$\supset$  \verb|\supset|	&$\subseteq$  \verb|\subseteq|	& $\supseteq$  \verb|\supseteq|	&$\in$  \verb|\in| \\
$\ni$  \verb|\ni|	 	&$\leq$  \verb|\leq|		& $\geq$  \verb|\geq|		 	&$\equiv$  \verb|\equiv| \\
$\sim$  \verb|\sim|		 &$\simeq$ \verb|\simeq| 	&$\approx$  \verb|\approx|		& $\cong$  \verb|\cong| \\		
$\neq$  \verb|\neq|		&$\doteq$  \verb|\doteq|	& $\perp$  \verb|\perp|	 	&$\mid$  \verb|\mid| \\
$\parallel$  \verb|\parallel|	&$\prec$  \verb|\prec|	& $\succ$  \verb|\succ| 		&$\preceq$  \verb|\preceq| \\
$\succeq$  \verb|\succeq|	&$\sqsubset$	\verb|\sqsubset|&$\sqsupset$ \verb|\sqpubset|	&$\sqsubseteq$ \verb|\sqsubseteq|		 \\ 
$\sqsupseteq$ \verb|\sqpubseteq|&$\smile$ \verb|\smile|	&$\frown$ \verb|\frown|		&$\notin$ \verb|\notin|  \\ 
$\models$ \verb|\models|	&$\asymp$ \verb|\asymp|	&$\bowtie$ \verb|\bowtie|		&$\vdash$ \verb|\vdash|  \\
$\dashv$ \verb|\dashv|	&$\leqq$ \verb|\leqq|		&$\geqq$ \verb|\geqq|		&$\lessgtr$ \verb|\lessgtr|  \\
$\leqslant$ \verb|\leqslant|	&$\geqslant$ \verb|\geqslant|&$\lll$ \verb|\lll|			&$\ggg$ \verb|\ggg|  \\
$\triangleq$ \verb|\triangleq|	&$\circeq$ \verb|\circeq|	&$\lesssim$ \verb|\lesssim|		&$\gtrsim$ \verb|\gtrsim|  \\
\hline
  \end{tabular*}
\end{table}

\begin{table}[h!]
  \begin{tabular*}{17cm}{@{\extracolsep{\fill}} ccc}
{\bf Arrows}					&					&					\\  \hline
$\leftarrow$  \verb|\leftarrow| 		&$\rightarrow$  \verb|\rightarrow|	& $\longleftarrow$  \verb|\longleftarrow| 	\\
$\longrightarrow$  \verb|\longrightarrow| 	&$\Leftarrow$  \verb|\Leftarrow|	& $\Rightarrow$  \verb|\Rightarrow| 	\\
$\Longleftarrow$  \verb|\Longleftarrow| 	&$\Longrightarrow$  \verb|\Longrightarrow|	& $\uparrow$  \verb|\uparrow| 	\\
$\downarrow$  \verb|\downarrow| 	&$\Uparrow$  \verb|\Uparrow|	& $\Downarrow$  \verb|\Downarrow| 	\\
$\leftrightarrow$  \verb|\leftrightarrow| 	&$\Leftrightarrow$  \verb|\Leftrightarrow|	& $\longleftrightarrow$  \verb|\longleftrightarrow| 	\\
$\Longleftrightarrow$  \verb|\Longleftrightarrow|	&$\updownarrow$  \verb|\updownarrow| &$\Updownarrow$  \verb|\Updownarrow| 						 \\ 
$\nearrow$  \verb|\nearrow| 		&$\searrow$  \verb|\searrow|	& $\swarrow$  \verb|\swarrow| 	\\
$\nwarrow$  \verb|\nwarrow| 	&$\mapsto$ \verb|\mapsto|		& $\longmapsto$ \verb|\longmapsto|		\\
$\leftharpoonup$  \verb|\leftharpoonup|&$\rightharpoonup$  \verb|\rightharpoonup|&$\leftharpoondown$ \verb|\leftharpoondown|	\\
$\rightharpoondown$ \verb|\rightharpoondown|&$\hookleftarrow$ \verb|\hookleftarrow|&$\hookrightarrow$ \verb|\hookrightarrow|	\\
$\rightleftharpoons$ \verb|\rightleftharpoons|&$\leadsto$ \verb|\leadsto|&$\Lsh$ \verb|\Lsh|	\\
$\Rsh$ \verb|\Rsh|	&$\curvearrowleft$ \verb|\curvearrowleft|	&$\curvearrowright$ \verb|\curvearrowright|	\\
\hline
  \end{tabular*}
\end{table}

\begin{table}[h!]
  \begin{tabular*}{17cm}{@{\extracolsep{\fill}} cccc}
{\bf Accents}			&				&					&\\  \hline
$\hat{x}$  \verb|\hat{x}|	&$\check{x}$  \verb|\check{x}| & $\dot{x}$  \verb|\dot{x}|	&$\breve{x}$  \verb|\breve{x}| \\
$\acute{x}$  \verb|\acute{x}|&$\ddot{x}$  \verb|\ddot{x}|& $\grave{x}$  \verb|\grave{x}|	&$\tilde{x}$  \verb|\tilde{x}| \\
$\mathring{x}$  \verb|\mathring{x}| 	&$\vec{x}$  \verb|\vec{x}|	& $\bar{x}$  \verb|\bar{x}| 	& 		\\ \hline
  \end{tabular*}
\end{table}

\begin{table}[h!]
  \begin{tabular*}{17cm}{@{\extracolsep{\fill}} cccc}
{\bf Dots}				&				&				&\\  \hline
$\ldots$  \verb|\ldots|		&$\cdots$  \verb|\cdots| & 		$\vdots$  \verb|\vdots|	&$\ddots$  \verb|\ddots| \\
\hline
  \end{tabular*}
\end{table}

\begin{table}[h!]
  \begin{tabular*}{17cm}{@{\extracolsep{\fill}} cccccc}
{\bf Delimiters}	&			&			&			&			&	\\  \hline
$|$  \verb+|+		&$\vert$  \verb|\vert| & $\|$  \verb+\|+	&$\Vert$  \verb|\Vert| &$\{$ \verb+\{+	&$\}$ \verb+\}+\\
$\langle$ \verb|\langle|&$\rangle$ \verb|\rangle|&$\lfloor$ \verb|\lfloor|&$\rfloor$ \verb|\rfloor|&$\lceil$ \verb|\lceil|&$\rceil$ \verb|\rceil|\\
$/$ \verb|/|	&$\backslash$ \verb|\backslash|&$[$ \verb|[|	&$]$ \verb|]|	&$\llcorner$ \verb|\llcornerl|&$\lrcorner$ \verb|\lrcorner|\\
$\ulcorner$ \verb|\ulcorner|	&$\urcorner$ \verb|\urcorner|	&		&		&		& 		\\
\hline
  \end{tabular*}
\end{table}

\begin{table}[h!]
  \begin{tabular*}{17cm}{@{\extracolsep{\fill}} cccc}
{\bf Mathematical Functions}	&				&					&\\  \hline
$\sin$  \verb|\sin|		&$\cos$  \verb|\cos|		& $\tan$  \verb|\tan|	 		&$\arcsin$  \verb|\arcsin| \\
$\arccos$  \verb|\arccos|	&$\arctan$  \verb|\arctan|	& $\sinh$  \verb|\sinh|		&$\cosh$  \verb|\cosh| \\
$\tanh$  \verb|\tanh|	 	&$\exp$  \verb|\exp|		& $\log$  \verb|\log|		 	&$\ln$  \verb|\ln| \\
$\sec$  \verb|\sec|		 &$\csc$ \verb|\csc| 		&$\cot$  \verb|\cot|			& $\max$  \verb|\max| \\		
$\min$  \verb|\min|		&$\dim$  \verb|\dim|		& $\inf$  \verb|\inf|	 		&$\lim$  \verb|\lim| \\
$\det$  \verb|\det|		&$\arg$  \verb|\arg|		& $\sup$  \verb|\sup| 			&$\deg$  \verb|\deg| \\
$\gcd$  \verb|\gcd|		&$\Pr$  \verb|\Pr|		&$\liminf$  \verb|\liminf| 		&$\limsup$  \verb|\limsup|	 \\ 
$\lg$  \verb|\lg|		&$\hom$  \verb|\hom|		&$\ker$  \verb|\ker|	 		&$\coth$  \verb|\coth| \\ \hline
  \end{tabular*}
\end{table}

\begin{table}[h!]
  \begin{tabular*}{17cm}{@{\extracolsep{\fill}} cccc}
{\bf Other Symbols}		&				&					&\\  \hline
$\aleph$  \verb|\aleph|	&$\prime$  \verb|\prime|	& $\forall$  \verb|\forall| 		&$\hbar$  \verb|\hbar| \\
$\emptyset$  \verb|\emptyset|&$\exists$  \verb|\exists|	& $\imath$  \verb|\imath|		&$\jmath$  \verb|\jmath| \\
$\nabla$  \verb|\nabla| 	&$\triangle$  \verb|\triangle|	& $\surd$  \verb|\surd|	 	&$\ell$  \verb|\ell| \\
$\partial$  \verb|\partial|	 &$\wp$ \verb|\wp| 		&$\Re$  \verb|\Re|			& $\Im$  \verb|\Im| \\		
$\angle$  \verb|\angle|	&$\mho$  \verb|\mho|		& $\infty$  \verb|\infty|	 	&$\Bbbk$  \verb+\Bbbk+ \\
$\sum$  \verb|\sum|		&$\int$  \verb|\int|		& $\oint$  \verb|\oint|			&$\prod$  \verb|\prod| \\
$\coprod$  \verb|\coprod|	&$\bigcap$  \verb|\bigcap|	& $\bigcup$  \verb|\bigcup| 		&$\bigodot$  \verb|\bigodot| \\ 
$\bigotimes$  \verb|\bigotimes|&$\bigoplus$  \verb|\bigoplus|& $\bigvee$  \verb|\bigvee| 	&$\bigwedge$  \verb|\bigwedge| \\ 
$\complement$  \verb|\complement|&$\nexists$  \verb|\nexists|& $\neg$  \verb|\neg|	 		&$\top$  \verb|\top| \\ 
$\bot$  \verb|\bot|		&$\Finv$ \verb|\Finv|		& $\varnothing$  \verb|\varnothing|	&$\Join$  \verb|\Join| \\
$\biguplus$  \verb|\biguplus|	&$\eth$ \verb|\eth|		& $\measuredangle$  \verb|\measuredangle|&$\sphericalangle$  \verb|\sphericalangle| \\
$\spadesuit$ \verb|\spadesuit|&$\heartsuit$ \verb|\heartsuit|& $\clubsuit$ \verb|\clubsuit|	&$\diamondsuit$ \verb|\diamondsuit| \\
$\sharp$ \verb|\sharp|	&$\flat$ \verb|\flat|		& $\natural$ \verb|\natural|		&$\square$ \verb|\square| \\
$\iint$ \verb|\iint|		&$\iiint$ \verb|\iiint|		& $\Game$ \verb|\Game|		&$\lozenge$ \verb|\lozenge| \\
$\circledS$ \verb|\circledS|	&$\backprime$ \verb|\backprime|	& $\bigstar$ \verb|\bigstar|	&$\hslash$ \verb|\hslash| \\
\hline
  \end{tabular*}
\end{table}

\begin{table}[h!]
  \begin{tabular*}{17cm}{@{\extracolsep{\fill}} lll}
{\bf Cross References}		&					&					\\  
{\sf Reference objects}		&{\sf Label commands}		&{\sf Reference Commands}		\\  \hline
Equations				&\verb+\label{eq:name}+		& \verb+\ref{eq:name}+ 	\\
Figures 				&\verb+\label{fig:name}+		& \verb+\ref{fig:name}+  	\\
Tables	 				&\verb+\label{tb:name}+		& \verb+\ref{tb:name}+ 	\\
Chapters 				&\verb+\label{ch:name}+		& \verb+\ref{ch:name}+ 	\\
Sections 				&\verb+\label{sec:name}+		& \verb+\ref{sec:name}+  	\\
Pages	 				&\verb+\label{page number}+	& \verb+\pageref{page number}+  	\\
Bibliography 				&\verb+\bibitem{name}+		& \verb+\cite{name}+	 	\\
\hline
  \end{tabular*}
	\caption{Referring directly to tables and figures requires captions.}
	\label{tb:cref}
\end{table}


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\bibitem{revtex}  REV{\TeX} Home Page,
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\bibitem{UCB} University of Colorado at Boulder, Department of Applied Mathematics,
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\bibitem{UK}The UK {\TeX} Archive,
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\end{thebibliography}



\end{document}
