Math Rendering Test
2026-03-25
Testing math and theorem environments.
Inline and Display Math
Inline: the fundamental theorem says $\int_a^b f’(x),dx = f(b) - f(a)$.
Display:
$$\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$$
With label and numbering:
$$E = mc^2 \label{eq:einstein} \tag{1}$$
Reference: see equation $\eqref{eq:einstein}$.
Theorem Environments
For any population protocol, $S = \sum x_i$ satisfies $S' = 0$ formally. Therefore for any $F$,
$$\frac{d}{dt}F(S) = F'(S) \cdot S' = 0.$$
Direct computation using the chain rule and the fact that $S' = r' + \sum x_i' = 0$ by construction of the PP.
A variable system has the one-marking property if each species $x_i$ is tracked by exactly one variable, giving $O(n)$ total variables.
For any non-affine univariate function $f$, the sum $\sum f(x_i)$ cannot be expressed as a function of $\sum x_i$ alone.
This is why the sqrt method fails: $\sum \sqrt{x_i} \neq g(\sum x_i)$ for any $g$.
Matrix
$$\begin{pmatrix} a_r \\ a_x \\ b_{rr} \\ b_{rx} \\ b_{xx} \end{pmatrix}' = M \begin{pmatrix} a_r \\ a_x \\ b_{rr} \\ b_{rx} \\ b_{xx} \end{pmatrix}$$
Macros
The natural numbers $\N$, the reals $\R$, expectation $\E[X]$, and $\norm{v} = \sqrt{\sum v_i^2}$.