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tt

This entry links every published equation using this exact notation. Its meaning may change between equations.

Meanings in context

Used in 200 equations

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),

How Multimodal Models Actually Handle Video, Audio, and Space · Equation 13

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

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Lseq=−∑t=1Tlog⁡pθ ⁣(zt∣z<t),zt∈{0,1,…,V−1}\mathcal{L}_{\text{seq}} = -\sum_{t=1}^{T} \log p_\theta\!\left(z_t \mid z_{<t}\right), \qquad z_t \in \{0, 1, \dots, V-1\}

Adapters, Native Pretraining, and Unified Tokens: The Main Multimodal Architectures, Compared · Equation 9

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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Ldiffusion=Ex0, ϵ∼N(0,I), t[ ∥ϵ−ϵθ(xt,t,c)∥2 ],xt=αˉt x0+1−αˉt ϵ\mathcal{L}_{\text{diffusion}} = \mathbb{E}_{x_0,\, \epsilon \sim \mathcal{N}(0, I),\, t} \Big[\, \big\| \epsilon - \epsilon_\theta(x_t, t, c) \big\|^2 \,\Big], \qquad x_t = \sqrt{\bar{\alpha}_t}\, x_0 + \sqrt{1 - \bar{\alpha}_t}\, \epsilon

Adapters, Native Pretraining, and Unified Tokens: The Main Multimodal Architectures, Compared · Equation 12

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

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∇θJ(θ)=Eπ ⁣[∑t=0T∇θlog⁡πθ(at∣st)(∑t′≥trt′)].\nabla_\theta J(\theta) = \mathbb{E}_\pi\!\left[\sum_{t=0}^{T} \nabla_\theta \log \pi_\theta(a_t \mid s_t) \left(\sum_{t' \ge t} r_{t'}\right)\right].

The Hardest Unsolved Problems in AI Agent Architecture · Equation 1

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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