Math rendering test

If you can read every equation below cleanly, the KaTeX integration is wired correctly. This page exercises every macro defined in src/lib/katex-macros.ts.

Continuous-time state space model

The continuous-time linear SSM:

\ddt \statevec(t) = \statemat \statevec(t) + \inputmat \bm{x}(t), \qquad \bm{y}(t) = \outputmat \statevec(t) + \feedmat \bm{x}(t)

Zero-order-hold discretization

With step size $\stepsize$ , the discretized SSM is:

\statevec_k = \discA \statevec_{k-1} + \discB \bm{x}_k, \qquad \discA = \exp(\stepsize \statemat)

Dimensions

A diagonal SSM has $\statedim$ state dimensions over $\seqlen$ time steps with input dimension $\inputdim$ .

Scan operator

The associative binary operator for diagonal SSMs combines pairs via:

(a_1, b_1) \scanop (a_2, b_2) = (a_1 \elemwise a_2,\; a_2 \elemwise b_1 + b_2)

Dynamical systems quantities

Stability is governed by the spectral radius $\spectralradius(\statemat) < 1$ , the largest Lyapunov exponent $\lyapexp_{\max}$ , and the monodromy matrix $\monodromy$ with Floquet multipliers $\floquet_i$ . The Jacobian $\jacobian$ measures local linearization.

Calculus

The total derivative $\ddt f$ and the partial derivative $\pderiv{f}{x}$ should both render with proper spacing.

Common sets and probability

The state lives in $\R^{\statedim}$ , complex eigenvalues in $\C$ , time indices in $\N$ , and shifts in $\Z$ . Expectations $\E[\bm{x}]$ and probabilities $\Prob(\statevec \in S)$ use the standard blackboard fonts.

Norms, inner products, absolute values

The training objective minimizes $\norm{\bm{y} - \hat{\bm{y}}}^2$ over $\bm{x}, \bm{y}$ pairs. Inner products $\ip{\bm{u}}{\bm{v}}$ and absolute values $\abs{x - x_0}$ render with paired delimiters.

Operators

Optimization searches via $\argmax_{\theta} \mathcal{L}(\theta)$ and $\argmin_{\theta} \mathcal{L}(\theta)$ . Matrix structure operators: $\diag(\bm{v})$ , $\tr(\statemat)$ , $\spec(\statemat)$ , $\rank(\statemat)$ .

Complexity

Naïve attention costs $\bigO{\seqlen^2 \statedim}$ . The associative scan reduces this to $\bigO{\seqlen \log \seqlen \cdot \statedim^2}$ .

If any equation above shows unrendered LaTeX source or a red error box, check that:

npm install completed without errors.
astro.config.mjs includes remarkPlugins: [remarkMath] and rehypePlugins: [[rehypeKatex, { macros: ssmMacros, ... }]].
src/layouts/Base.astro imports 'katex/dist/katex.min.css'.