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jj

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

Used in 58 equations

uc2(y)=∑i=1N(∂f∂xi)2u2(xi)  +  2∑i=1N−1∑j=i+1N∂f∂xi∂f∂xj u(xi,xj)u_c^2(y) = \sum_{i=1}^{N} \left(\frac{\partial f}{\partial x_i}\right)^2 u^2(x_i) \;+\; 2\sum_{i=1}^{N-1}\sum_{j=i+1}^{N} \frac{\partial f}{\partial x_i}\frac{\partial f}{\partial x_j}\, u(x_i, x_j)

How Scientific Instruments and Metrology Actually Work · Equation 4

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

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RMSNorm(x)j=xjRMS(x) gj,RMS(x)=1d∑k=1dxk2+ϵ.\mathrm{RMSNorm}(x)_j = \frac{x_j}{\mathrm{RMS}(x)}\, g_j, \qquad \mathrm{RMS}(x) = \sqrt{\frac{1}{d}\sum_{k=1}^{d} x_k^2 + \epsilon}.

How Llama's Architecture Actually Works, Generation by Generation · Equation 19

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

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L=−12B∑i=1B[log⁡exp⁡(⟨ui,vi⟩/τ)∑j=1Bexp⁡(⟨ui,vj⟩/τ)+log⁡exp⁡(⟨ui,vi⟩/τ)∑j=1Bexp⁡(⟨uj,vi⟩/τ)],\mathcal{L} = -\frac{1}{2B}\sum_{i=1}^{B}\left[\log\frac{\exp(\langle u_i, v_i\rangle/\tau)}{\sum_{j=1}^{B}\exp(\langle u_i, v_j\rangle/\tau)} + \log\frac{\exp(\langle u_i, v_i\rangle/\tau)}{\sum_{j=1}^{B}\exp(\langle u_j, v_i\rangle/\tau)}\right],

A History of Multimodal AI · Equation 5

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

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L=−1B∑i=1Blog⁡exp⁡(⟨ui,vi⟩/τ)∑j=1Bexp⁡(⟨ui,vj⟩/τ),\mathcal{L} = -\frac{1}{B} \sum_{i=1}^{B} \log \frac{\exp(\langle u_i, v_i \rangle / \tau)}{\sum_{j=1}^{B} \exp(\langle u_i, v_j \rangle / \tau)},

One Model, Many Modalities: What Multimodal Systems Actually Share · Equation 6

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

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ci=∑j=1Txαijhj,αij=exp⁡(eij)∑k=1Txexp⁡(eik).c_i = \sum_{j=1}^{T_x} \alpha_{ij} h_j, \qquad \alpha_{ij} = \frac{\exp(e_{ij})}{\sum_{k=1}^{T_x} \exp(e_{ik})}.

From n-Grams to Reasoning Models: A Technical History of the Language Model · Equation 5

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

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