Live mathematical reference

Mathematical compendium

A live index of the equations in published articles. Each entry leads to its article, equation guide, and the terms explained there. New or edited published articles appear automatically.

1680 equations across 1484 symbols.

σ^  =  Nobs−N^bkgA ε L\hat{\sigma} \;=\; \frac{N_{\mathrm{obs}} - \hat{N}_{\mathrm{bkg}}}{\mathcal{A}\,\varepsilon\,\mathcal{L}}

1 published occurrence

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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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p^k+1=F(D0∪S1∪⋯∪Sk)\hat{p}_{k+1} = \mathcal{F}\left( D_0 \cup S_1 \cup \cdots \cup S_k \right)

1 published occurrence

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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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s^=1n∑i=1n1 ⁣[ success on xi ],xi∼Dbench\hat{s} = \frac{1}{n}\sum_{i=1}^{n} \mathbf{1}\!\left[\,\text{success on } x_i\,\right], \qquad x_i \sim \mathcal{D}_{\mathrm{bench}}

1 published occurrence

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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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⟨exp⁡(−ΔstotkB)⟩=1\left\langle \exp\left(-\frac{\Delta s_{\mathrm{tot}}}{k_{\mathrm B}}\right) \right\rangle = 1

1 published occurrence

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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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⟨e−βW⟩=e−βΔF,β=1kBT\left\langle e^{-\beta W}\right\rangle = e^{-\beta\Delta F}, \qquad \beta = \frac{1}{k_{\mathrm B}T}

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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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lim⁡d→∞E ⁣[Dmax⁡k−Dmin⁡kd1/k−1/2]=C⋅1(k+1)1/k12k+1\lim_{d \to \infty} \mathbb{E}\!\left[ \frac{D_{\max}^{k} - D_{\min}^{k}}{d^{1/k - 1/2}} \right] = C \cdot \frac{1}{(k+1)^{1/k}} \sqrt{\frac{1}{2k+1}}

1 published occurrence

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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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lim⁡m→∞var⁡ ⁣(dm(Pm,Qm)pE[dm(Pm,Qm)p])=0    ⟹    lim⁡m→∞Pr⁡ ⁣[Dmax⁡(m)≤(1+ε) Dmin⁡(m)]=1\lim_{m \to \infty} \operatorname{var}\!\left( \frac{d_m(P_m, Q_m)^p}{\mathbb{E}\left[ d_m(P_m, Q_m)^p \right]} \right) = 0 \;\;\Longrightarrow\;\; \lim_{m \to \infty} \Pr\!\left[ D_{\max}^{(m)} \le (1 + \varepsilon)\, D_{\min}^{(m)} \right] = 1

1 published occurrence

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This equation states a bound: one expression must stay on the indicated side of the other under the article’s assumptions.

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E[rθ(x,y)]−β E[DKL ⁣(πRL ∥ πSFT)]+γ E[log⁡πRL(x)]\mathbb{E}\big[r_\theta(x,y)\big] - \beta\,\mathbb{E}\big[D_{\mathrm{KL}}\!\left(\pi^{\mathrm{RL}} \,\|\, \pi^{\mathrm{SFT}}\right)\big] + \gamma\,\mathbb{E}\big[\log \pi^{\mathrm{RL}}(x)\big]

1 published occurrence

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This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

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