testing

this is some inline math $\textstyle R^{n}\Rightarrow R^{n}$ to test things out.

$\int _{i=0}^{20}\sin \theta ^{\phi }\ dx$

$x\otimes y={\tilde {y}}{\tilde {x}}+x\cdot y\mathbf {1}$

${\begin{array}{lcr}{\mbox{First number}}&x&8\\{\mbox{Second number}}&y&15\\{\mbox{Sum}}&x+y&23\\{\mbox{Difference}}&x-y&-7\\{\mbox{Product}}&xy&120\end{array}}$

$|x|=\left\{{\begin{array}{ll}\;x&{\mbox{if}}\ x\geq 0;\\-x&{\mbox{if}}\ x<0.\end{array}}\right.$

$x=3$