Kanta Hayashi
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Research//2 min readTest post

Give the equations some room

A math specimen: inline symbols, matrices, aligned equations, and one deliberately wide expression.

Kanta Hayashi

This is a typesetting test, not a research result. It checks whether the page can carry a mathematical explanation without making the surrounding prose difficult to read.

Inline notation should feel like part of a sentence

Let x∈Rdx\in\mathbb{R}^{d} and let W∈Rm×dW\in\mathbb{R}^{m\times d}. A linear map sends xx to WxWx. The symbols should sit on the baseline, and the sentence should keep its rhythm.

For a categorical distribution, pk≥0p_k\geq 0 and ∑k=1Kpk=1\sum_{k=1}^{K}p_k=1. Here the subscript, superscript, and Greek symbols α,β,θ\alpha,\beta,\theta exercise different glyphs.

Several lines, one argument

A stable way to write a normalized exponential is

m=max⁡jzj,log⁡Z=m+log⁡∑jexp⁡(zj−m),pi=exp⁡(zi−log⁡Z).\begin{aligned} m &= \max_j z_j,\\ \log Z &= m + \log\sum_j\exp(z_j-m),\\ p_i &= \exp(z_i-\log Z). \end{aligned}

The layout should preserve alignment and give the display a little space. Its background should not turn white when the rest of the page switches to dark mode.

A matrix and a piecewise expression

A=[10−1020−101],f(x)={x2,x≥0,0,x<0.A=\begin{bmatrix} 1 & 0 & -1\\ 0 & 2 & 0\\ -1 & 0 & 1 \end{bmatrix},\qquad f(x)=\begin{cases} x^2,&x\geq 0,\\ 0,&x<0. \end{cases}

This tests delimiter sizing and negative signs. It makes no statement about a trained model.

A deliberately wide equation

The following display is intentionally too wide for a phone. Only the equation should scroll; the whole page should not.

Lillustrative(θ)=1N∑i=1N−log⁡pθ(yi∣xi)⏟example data term+λ1∥θ∥22⏟example regularizer+λ2∑j=1d(θj−θ‾)2⏟another illustrative term+λ3∑k=1K(qk−1K)2⏟deliberately long final term.\mathcal{L}_{\mathrm{illustrative}}(\theta) =\underbrace{\frac{1}{N}\sum_{i=1}^{N}-\log p_{\theta}(y_i\mid x_i)}_{\text{example data term}} +\underbrace{\lambda_1\lVert\theta\rVert_2^2}_{\text{example regularizer}} +\underbrace{\lambda_2\sum_{j=1}^{d}\left(\theta_j-\overline{\theta}\right)^2}_{\text{another illustrative term}} +\underbrace{\lambda_3\sum_{k=1}^{K}\left(q_k-\frac{1}{K}\right)^2}_{\text{deliberately long final term}}.

This expression exists only to test overflow. It is not a proposed loss function or a recommendation.

Back to the prose

The paragraph after the equation should return to the same measure and font size. On a small screen, scroll the display sideways, then continue reading. On a desktop, enlarge the page to 200% and check that the title, table of contents, and formulas still fit.

The research specimen mixes prose, equations, and data. The Japanese specimen checks a different rhythm.