Testing MathML

Inline math: $\sin(x_n^2)$ . And displayed math:

$f(\epsilon) = \frac{1}{1 + \exp\left(\frac{\varepsilon}{k_\text{B}T}\right)}$ $N = \frac{\text{number of apples}}{7}$ $\mathbf{M} = \left(\begin{matrix}a&b\\c&d\end{matrix}\right)$

We have $|\mathbf{M}| = ad - bc$ .

$\int_0^1 x^n dx = \frac{1}{n + 1}$ $\sum_{n=1}^m n = \frac{m(m+1)}{2}$

tilde: $\tilde{n}$
hat: $\hat{H}$
bar: $\bar{v}$

Quantum mechanics:

$-\frac{1}{2}\nabla^2 \psi + v \psi = \varepsilon \psi$

Math split over two lines:

$g(\alpha) = & (1 + \alpha + \alpha^2 + \alpha^3 + \alpha^4\\ & + \alpha^5)$ $f(x) = \left\{ \begin{matrix} 1 - x, & x < 1 \\ 0, & x > 1 \end{matrix}\right.$