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.

ci=pin(τ+δ+(i−1)σ)+pout oˉc_i = p_{\text{in}}\big(\tau + \delta + (i-1)\sigma\big) + p_{\text{out}}\,\bar{o}

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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Cn=∑i=1nci=n pin(τ+δ)+pin σ n(n−1)2+n pout oˉC_n = \sum_{i=1}^{n} c_i = n\,p_{\text{in}}(\tau+\delta) + p_{\text{in}}\,\sigma\,\frac{n(n-1)}{2} + n\,p_{\text{out}}\,\bar{o}

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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Cncached≈n pout oˉ+n pin w σ+pin γ σ n(n−1)2C_n^{\text{cached}} \approx n\,p_{\text{out}}\,\bar{o} + n\,p_{\text{in}}\,w\,\sigma + p_{\text{in}}\,\gamma\,\sigma\,\frac{n(n-1)}{2}

1 published occurrence

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This equation gives an approximation: it relates the quantities while allowing an approximation.

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C(r)=∫tntfT(t) σ(r(t)) c(r(t),d) dt,T(t)=exp⁡ ⁣(−∫tntσ(r(s)) ds)C(\mathbf{r}) = \int_{t_n}^{t_f} T(t)\, \sigma(\mathbf{r}(t))\, \mathbf{c}(\mathbf{r}(t), \mathbf{d})\, dt, \qquad T(t) = \exp\!\left(-\int_{t_n}^{t} \sigma(\mathbf{r}(s))\, ds\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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C(t)=1 ⁣[U(t)≥u∗]⋅1 ⁣[R(t)≥r∗]C(t) = \mathbb{1}\!\left[U(t) \ge u^{*}\right] \cdot \mathbb{1}\!\left[R(t) \ge r^{*}\right]

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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D(t)=1 ⁣[P(t)≥p∗]⋅1 ⁣[L(t)≥ℓ∗]D(t) = \mathbb{1}\!\left[P(t) \ge p^{*}\right] \cdot \mathbb{1}\!\left[L(t) \ge \ell^{*}\right]

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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G∈{1,2,…,H},G=H⇒multi-head,G=1⇒multi-query,1<G<H⇒grouped-queryG \in \{1, 2, \dots, H\}, \qquad G = H \Rightarrow \text{multi-head}, \quad G = 1 \Rightarrow \text{multi-query}, \quad 1 < G < H \Rightarrow \text{grouped-query}

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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G=(V,E),v→v′ fires iff γ(v,v′)=1G = (V, E), \qquad v \to v' \text{ fires iff } \gamma(v, v') = 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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G(t)=C(t)−V(t),dGdt=dCdt−dVdtG(t) = C(t) - V(t), \qquad \frac{\mathrm{d}G}{\mathrm{d}t} = \frac{\mathrm{d}C}{\mathrm{d}t} - \frac{\mathrm{d}V}{\mathrm{d}t}

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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