Math Jax Example

In $N$-dimensional simplex noise, the squared kernel summation radius $r^2$ is $\frac{1}{2}$ for all values of $N$. This is because the edge length of the $N$-simplex $s = \sqrt{\frac{N}{N+1}}$ divides out of the $N$-simplex height $h = \sqrt{\frac{N+1}{2N}}$. The kernel summation radius $r$ is equal to the $N$-simplex height $h$.

Other example: $ \psi\rangle$

$$r = h = \sqrt{\frac{1}{2}} = \sqrt{\frac{N}{N+1}} \sqrt{\frac{N+1}{2N}}$$

Let’s test some inline math $x$, $y$, $x_1$, $y_1$.

Now a inline math with special character: $|\psi\rangle$, $x’$, $x^* $.

Test a display math: $|\psi_1\rangle = a|0\rangle + b|1\rangle$ Is it O.K.?

Test a display math with equation number: \begin{equation} |\psi_1\rangle = a|0\rangle + b|1\rangle \end{equation} Is it O.K.?

Test a display math with equation number: $\begin{align} |\psi_1\rangle &= a|0\rangle + b|1\rangle \\\\ |\psi_2\rangle &= c|0\rangle + d|1\rangle \end{align}$ Is it O.K.?

And test a display math without equation number: $\begin{align} |\psi_1\rangle &= a|0\rangle + b|1\rangle \\\\ |\psi_2\rangle &= c|0\rangle + d|1\rangle \end{align}$ Is it O.K.?

Test a display math with equation number: \begin{align} |\psi_1\rangle &= a|0\rangle + b|1\rangle \
|\psi_2\rangle &= c|0\rangle + d|1\rangle \end{align} Is it O.K.?

And test a display math without equaltion number: \begin{align} |\psi_1\rangle &= a|0\rangle + b|1\rangle \
|\psi_2\rangle &= c|0\rangle + d|1\rangle \end{align} Is it O.K.?