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\usepackage{amsmath,amsthm,amsfonts,amssymb,latexsym,hyperref}
\usepackage{graphics,graphicx,color,array}
\usepackage[all]{xy}
\usepackage{afterpage}
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\newtheorem{theorem} {Theorem}

\newtheorem{example} {Example}

\newtheorem{step} {Step}

\newtheorem{claim} {Claim}

\newtheorem{definition} {Definition}

\newtheorem{corollary} {Corollary}





\begin{document}


Equation Arrays: it's automatically in the mathematics environment; you don't need to use \$ signs


\begin{eqnarray*}
\sum_{i=1}^{k+1} i^2 & = & \sum_{i=1}^{k} i^2 + (k+1)^2 \\
  & = & \frac{k(k+1)(2k+1)}{6} + (k+1)^2 \\ 
  & = & \mbox{etc. } \ldots.
\end{eqnarray*}

\


Following are examples of matrices; the \{cccc\}:  is telling the computer to center the entry in the cells, there are four c's because there are (at least) four columns.  The other possibilities are left \{llll\} and right \{rrrr\}.  You can also mix them: \{clrc\}.

Theorem.


$$AB = \left [
\begin{array}{cccc}
 a_{1,1} & a_{1,2} & ... & a_{1,n}
\\ a_{2,1} & a_{2,2} & ... & a_{2,n}
\\ \vdots
\\ a_{m,1} & a_{m,2} & ... & a_{m,n}
\end{array} \right ]
\left [
\begin{array}{cccc}
 b_{1,1} & b_{1,2} & ... & b_{1,k}
\\ b_{2,1} & b_{2,2} & ... & b_{2,k}
\\ \vdots
\\ b_{n,1} & b_{n,2} & ... & a_{n,k}
\end{array} \right ]=
\left [
\begin{array}{cccc}
 c_{1,1} & c_{1,2} & ... & c_{1,k}
\\ c_{2,1} & c_{2,2} & ... & c_{2,k}
\\ \vdots
\\ c_{n,1} & \ & ... & c_{m,k}
\end{array} \right ]= C$$

\

$$f'(P)(\vec{v}) = \left [
\begin{array}{ccccc}
 D_1(f_1)(P) & D_2(f_1)(P) & ... & D_n(f_1)(P)
\\ D_1(f_2)(P) & D_2(f_2)(P) & ... & D_n(f_2)(P)
\\ \vdots
\\ D_1(f_m)(P) & D_2(f_m)(P) & ... & D_n(f_m)(P)
\end{array} \right ]
\left [
\begin{array}{ccccc}
 v_1
\\ v_2
\\ \vdots
\\ v_n
\end{array} \right ]$$


\


$$
\begin{array}{cc}
 D_1(f_1)(P) & D_2(f_1)(P)
\\ D_1(f_2)(P) & D_2(f_2)(P)
\\ \vdots
\\ D_1(f_m)(P) & D_2(f_m)(P)
\end{array}
\begin{array}{ccccc}
 v_1 & 2
\\ v_2 & 3
\\ \vdots
\\ v_n & 4
\end{array}$$

\

$$
\begin{array}{cc}
 D_1(f_1)(P)
\\ D_1(f_2)(P)
\\ \vdots
\\ D_1(f_m)(P)
\end{array} \ \
\begin{array}{cc}
 v_1
\\ v_2
\\ \vdots
\\ v_n
\end{array}$$

\

Function definitions:

\

$$h(x) = \left \{
\begin{array}{cllc}
f(x) & \mbox{if } a\le x< b \\
 d & \mbox{if } x=b \\
g(x) & \mbox{if } b < x \le c. \\
\end{array} \right . $$

\

$$f = \left \{
\begin{array}{cccc}
y_1 + \sin x
\\ y_2 + \cos x
\\ y_3 + \tan x
\\ y_4
\end{array} \right . $$

\


$f = \left \{
\begin{array}{cccc}
y_1 + \sin x
\\ y_2 + \cos x
\\ y_3 + \tan x
\\ y_4
\end{array} \right .$

\

\


Function definition:
$f = \left \{
\begin{array}{cccc}
y_1 + \sin x
\\ y_2 + \cos x
\\ y_3 + \tan x
\\ y_4
\\ y_5 + e^x
\end{array} \right .$

\
Commuting diagram:
$$\begin{array}{ccc}
x_{i-1} &  \begin{array}{c} f_{i-1} \\ {\longleftarrow} \end{array} & x_i \\
\downarrow & \circlearrowleft & \downarrow \\
p_{i-1} & \begin{array}{c} g_{i-1} \\ {\longleftarrow} \end{array}  & p_i
\end{array}$$

\begin{center}
\begin{displaymath}
\xymatrix{
\hat M \ar@{->}[r]^{\hat h }\ar@{->}[d]^{\pi}& \hat M \ar@{->}[d]^{\pi}\\
\hat M / G \ar@{->}[r]^{h}& \hat M / G
}
\end{displaymath}
\end{center}

\

\begin{center}
\begin{displaymath}
\xymatrix{
\hat M \ar@{<-}[r]^{\hat h }\ar@{->}[d]^{\pi}& \hat M \ar@{->}[d]^{\pi}\\
\hat M / G \ar@{<-}[r]^{h}& \hat M / G
}
\end{displaymath}
\end{center}

\

\begin{center}
\begin{displaymath}
\xymatrix{
\hat M \ar@{<-}[r]^{\hat h }\ar@{->}[d]^{\pi}& \hat M \ar@{->}[d]^{\pi}\\
\hat M \ar@{<-}[r]^{\hat h }\ar@{->}[d]^{\pi}& \hat M \ar@{->}[d]^{\pi}\\
\hat M / G = M \ar@{<-}[r]^{h}& M = \hat M / G
}
\end{displaymath}
\end{center}

\


Following is the format for a table (e.g. , multiplication tables, logic tables).

\
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|c|c||}
\hline
$D_2$ & $e$ &  $\rho_1$ &  $\rho_2$ &  $\rho_3$ & $\mu_1$ &  $\mu_2$ &  $\delta_1$ &  $\delta_2$   \\
\hline
\hline
$e$ & $e$ &  $\rho_1$ &  $\rho_2$ &  $\rho_3$ & $\mu_1$ &  $\mu_2$ &  $\delta_1$ &  $\delta_2$   \\
\hline
$\rho_1$ & $\rho_1$&  $\rho_2$ &  $\rho_3$ &  $e$ & $\delta_1$ &  $\delta_2$ &  $\mu_2$ &  $\mu_1$   \\
\hline
$\rho_2$ &  $\rho_2$ &  $\rho_3$ &  $e$ &  $\rho_1$ & $\mu_2$ &  $\mu_1$ &  $\delta_2$ &  $\delta_1$   \\
\hline
$\rho_3$ & $\rho_3$&  $e$ &  $\rho_1$ &  $\rho_2$ & $\delta_2$ &  $\delta_1$ &  $\mu_1$ &  $\mu_2$   \\
\hline
$\mu_1$ & $\mu_1$ &  $\delta_2$ &  $\mu_2$ &  $\delta_1$ & $e$ & $\rho_2$&  $\rho_3$ &  $\rho_1$ \\
\hline
$\mu_2$ & $\mu_2$ &  $\delta_1$ &  $\mu_1$ &  $\delta_2$ & $\rho_2$ & $e$&  $\rho_1$ &  $\rho_3$ \\
\hline
$\delta_1$ & $\delta_1$ &  $\mu_1$ &  $\delta_2$ &  $\mu_2$ & $\rho_1$ & $\rho_3$ &  $e$ &  $\rho_2$ \\
\hline
$\delta_2$ & $\delta_2$ &  $\mu_2$ &  $\delta_1$ &  $\mu_1$ & $\rho_3$ & $\rho_1$ &  $\rho_2$ &  $e$ \\
\hline
\hline
\end{tabular}
\end{center}

\

Commuting diagrams:

\begin{center}
\begin{displaymath}
\xymatrix{
\hat M \ar@{<-}[r]^{\hat h }\ar@{->}[d]^{\pi}& \hat M \ar@{->}[d]^{\pi}\\
\hat M / G = M \ar@{<-}[r]^{h}& M = \hat M / G
}
\end{displaymath}
\end{center}


\

\end{document}