Math

This site supports LaTeX math via KaTeX and numbered math environment components for academic-style content.

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Inline and Display Math

Use single dollar signs for inline math: $E = mc^2$ renders as $E = mc^2$.

Use double dollar signs for display math:

$$\nabla_\theta J(\theta) = \mathbb{E}_{\pi_\theta} \left[ \sum_{t=0}^{T} \nabla_\theta \log \pi_\theta(a_t | s_t) \cdot G_t \right]$$

See the full list of supported functions at KaTeX docs.

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Numbered Equations

The <Equation> component renders a centered, numbered equation:

$L(\theta) = -\sum_{i=1}^{N} y_i \log \hat{y}_i$

$\text{softmax}(z_i) = \frac{e^{z_i}}{\sum_{j} e^{z_j}}$

Reference them inline: the cross-entropy loss uses the softmax .

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Definitions, Theorems, and Proofs

A function $f: \mathbb{R}^n \to \mathbb{R}$ is **convex** if for all $x, y \in \text{dom}(f)$ and $\lambda \in [0, 1]$:

$f(\lambda x + (1-\lambda) y) \leq \lambda f(x) + (1-\lambda) f(y)$

If $f$ is a convex function (see [](#def-convex)) and $X$ is a random variable, then:

$f(\mathbb{E}[X]) \leq \mathbb{E}[f(X)]$

By [](#def-convex), for any convex combination we have $f(\lambda x + (1-\lambda)y) \leq \lambda f(x) + (1-\lambda)f(y)$. Extending by induction to $n$ points with weights $\lambda_i \geq 0, \sum \lambda_i = 1$ gives $f\!\left(\sum_i \lambda_i x_i\right) \leq \sum_i \lambda_i f(x_i)$. Taking the limit to a continuous distribution yields the result.

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All Available Environments

All environments are auto-numbered and cross-referenceable via id. Use name for an optional label.

Component	Color	Use case	Syntax example
<Theorem>	Blue-gray	Main results	<Theorem id="thm-main" name="Main Result">...</Theorem>
<Lemma>	Blue-gray	Supporting results	<Lemma id="lem-bound">...</Lemma>
<Proposition>	Blue-gray	Intermediate claims	<Proposition id="prop-equiv">...</Proposition>
<Corollary>	Blue-gray	Direct consequences	<Corollary id="cor-rate">...</Corollary>
<Definition>	Green	Formal definitions	<Definition id="def-foo" name="Foo">...</Definition>
<Remark>	Yellow	Commentary and observations	<Remark id="rmk-note">...</Remark>
<Example>	Purple	Worked examples	<Example id="ex-demo">...</Example>
<Proof>	Left border	Proof with QED symbol	<Proof>...</Proof>
<Equation>	—	Centered numbered equation	<Equation id="eq-loss">$$...$$</Equation>
<Code>	—	Captioned code block	<Code id="code-foo" caption="...">...</Code>
<Algorithm>	Bordered	Pseudocode with caption	<Algorithm id="alg-foo" caption="...">...</Algorithm>