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Science, DecodedContext Jamming

Equation Inspector

Tap a symbol. Read the equation.

12 equations from physics, biology, information theory and machine learning. Open one, tap any symbol to see its job there, then reveal the readings from the gist to the expert level.

L(N)=L∞+A N−α{L(N)} = {L_{\infty}} + {A}\,{N}^{{-\alpha}}

An illustrative neural scaling law

Empirical regularityAI / MLMathematicsComplexity
F⃗=m a⃗{\vec{F}} = {m}\,{\vec{a}}

Newton’s second law

EstablishedPhysics
E=m c2{E} = {m}\,{c^{2}}

Mass–energy equivalence

EstablishedPhysics
λ=cf{\lambda} = \frac{{c}}{{f}}

Wavelength, speed and frequency

EstablishedPhysics
H=−∑ipi log⁡pi{H} = {-}{\sum_{i}} {p_i}\,{\log p_i}

Shannon entropy

EstablishedMathematicsAI / MLBiologyPhysics
H(p,q)=−∑ipi log⁡qi{H(p,q)} = -{\sum_{i}} {p_i}\,{\log q_i}

Cross-entropy (the language-model loss)

EstablishedAI / MLMathematics
θt+1=θt−η ∇θL(θt){\theta_{t+1}} = {\theta_t} - {\eta}\,{\nabla_\theta} {L(\theta_t)}

Gradient descent update

Mathematical modelAI / MLMathematics
Attention(Q,K,V)=softmax ⁣(QKTdk)V\mathrm{Attention}({Q},{K},{V}) = {\mathrm{softmax}}\!\left(\frac{{Q K^{T}}}{{\sqrt{d_k}}}\right){V}

Scaled dot-product attention

Mathematical modelAI / MLMathematics
P(H∣E)=P(E∣H) P(H)P(E){P(H \mid E)} = \frac{{P(E \mid H)}\,{P(H)}}{{P(E)}}

Bayes’ theorem

EstablishedMathematicsEpistemicsBiologyAI / ML
pi=ezi/T∑jezj/T{p_i} = \frac{{e^{z_i / T}}}{{\sum_{j} e^{z_j / T}}}

Softmax with temperature

EstablishedAI / MLMathematics
pi=e−Ei/kBTZ{p_i} = \frac{{e^{-E_i / k_B T}}}{{Z}}

The Boltzmann distribution

EstablishedPhysicsAI / MLMathematics
B∝M3/4{B} {\propto} {M}^{{3/4}}

Kleiber’s law (metabolic scaling)

Empirical regularityBiologyComplexity