History for LatexWikiTests

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Some tests and demos of LatexWiki
LatexWiki supports dollar sign type math mode expressions like $\alpha\in\mathcal{C}$.  You can use individual letters like $a$, $x$, and $\alpha$, $\beta$, $\omega$, $\xi$, and also $\phi$ and $\varphi$.  
You can have inline formulas like $\sum_{n=0}^{\infty}\frac{(-1)^n}{n}$ or in display mode like so:
\[ \sum_{n=1}^{\infty}\frac{(-1)^n}{n} \]
lets try $H^2 \rightarrow H^3$
We can use multiline \begin \end type constructs:
\begin{align*}
   f(\overline{E}) & = \bigcup_{\lambda\in\Lambda}f|_{A_{\lambda}}(\overline{E\cap A_{\lambda}})\\
    &\subseteq \bigcup_{\lambda\in\Lambda}\overline{f|_{A_{\lambda}}(E\cap A_{\lambda})}\\
   &\subseteq \overline{f(E)}
\end{align*}
Here is an example of dollar signs inside a \begin. It appears to work.
\begin{align*} \left( x^{-1} y x \right)^n &= x^{-1} y^n x && \text{ This is simply because the function $y \mapsto x^{-1} y x$ is a homomorphism. } 
\end{align*}
Here's a definition of the familiar factorial function, using the cases environment inside an equation environment:
\begin{equation*}
  f(n) = \begin{cases}
      1 &\text{ if n is 1}\\
      nf(n-1) &\text{ otherwise.}
 \end{cases}
\end{equation*}
Kyle's going to play some too $e^{i \pi} + 1 = 0$. 
Here's a commutative diagram, in the style of Josh Small, showing a map from Truck to Shoe thru $A\otimes B$:
\begin{equation*}
 \mbox{\xymatrix{
          \text{TRUCK}\ar@{-->}[dr]_{g\circ f}\ar[r]^f& A\otimes B
              \ar[d]^g\\
          &\text{SHOE}}} 
\end{equation*}
Other tests go here.
\begin{equation*}
\prod_{-\infty}^{\infty}\left(\frac{\displaystyle \sum_{i=0}^{\infty} i^k \frac{k^2 - i^\alpha}{\alpha^k - \pi^i }}{\displaystyle \sum_{j=1}^{\infty}\sqrt{j+\sqrt{k+\alpha^k}}}\right)
\end{equation*}
\begin{equation*}
 \mbox{\xymatrix{
           X \ar@{-->}[dr]_{g\circ f}\ar[r]^f& A\otimes B
              \ar[d]^g\\
          & Y}} 
\end{equation*}
\begin{equation*}
\prod\limits_{k=-\infty}^{\infty} 
\left( \frac{\sum\limits_{i=0}^{\infty} i^k 
             \frac{k^2 - i^\alpha}{\alpha^k - \pi^i }}
            {\sum\limits_{j=1}^{\infty} \sqrt{j+\sqrt{k+\alpha^k}}} 
\right)
\end{equation*}
\begin{equation*}
\cfrac{bla}{\cfrac{ok}{\alpha}}
\end{equation*}