Foundations

Different subsurface density structures can yield the same external gravitational field.

For stationary homogeneous material with constant positiveρ,c_p andk, no internal heat production and one-dimensional conduction, ρc_pT_t=kT_zz, with thermal diffusivityκ=k/(ρc_p).

With depthz positive downward, constant conductivityk and uniform volumetric heat productionH, a stationary conductive slab obeys kT_zz+H=0.

For z>0 initially at uniform T_i and a surface held at T_s for t>0, constant-κ conduction gives T(z,t)=T_s+(T_i−T_s)erf[z/(2√(κt))].

Passive sensors receive externally supplied or naturally emitted radiation; active radar/lidar supplies illumination and measures its return.

For the specified Landsat Level-1 product, metadata supplies band-specific rescaling L_λ=M_LQ_cal+A_L and uncorrected planetary reflectanceρ′=M_ρQ_cal+A_ρ.

A Landsat thermal-band TOA radianceL_λ>0 maps to product brightness temperature T_B=K₂/ln(K₁/L_λ+1), using its band-specific metadata constants.

For compatible near-infrared and red surface-reflectance bands, NDVI=(ρ_NIR−ρ_R)/(ρ_NIR+ρ_R).

For a monostatic pulse with corrected round-trip delayτ and constant propagation group speedv_g, one-way slant range is R=v_gτ/2.

Ocean-colour band-ratio models estimate near-surface chlorophyll-a from water-leaving remote-sensing reflectances.

LetΣ be the symmetric compression-positive stress tensor, the negative of tensile-positive Cauchy stress.

In an Andersonian idealization, one principal stress axis is vertical and two are horizontal.

In a quasistatic elastic fracture model, crack extension releases mechanical potential energy at rateG=−∂Π/∂A per projected crack area.

A weak-field linear isotropic rock model separates magnetizationM=M_r+χH into retained remanence and an induced response to localH.

The nerve of an ordinary category has n-simplices given by chains of n composable arrows; inner horns have unique fillers encoding composition.

For i u-t=−Delta-u−lambda|u|^alpha u on R-n, real lambda and 0<alpha<4/n, every L2 initial datum has a unique local mild solution in C([0,T];L2) intersected with L-q(0,T;L^(alpha+2)), where q=4(alpha+2)/(n alpha).

A randomized mechanism is (epsilon,delta)-differentially private if every neighboring dataset pair and every output event obey Pr[M(D) in S]≤exp(epsilon)Pr[M(D-prime) in S]+delta.

For a vector query with global L1 sensitivity Delta-1, adding independent Laplace noise of scale Delta-1/epsilon to each coordinate gives epsilon-DP under the chosen adjacency.

For order alpha>1, RDP bounds the Rényi divergence between mechanism outputs on every adjacent dataset pair by epsilon-alpha.

A norm-bounded adversarial evaluation asks whether an attacker can change a model's prediction using perturbations within a specified set around an input.

Gaussian smoothing defines a classifier by the most probable prediction of a base classifier under N(0,sigma²I) noise.

Scaled dot-product attention computes softmax(QK-transpose/sqrt(d-k))V, allowing token representations to combine context through learned query/key/value projections.

A DDPM adds Gaussian noise in a forward Markov chain and learns a parameterized reverse process to generate samples.

An autoregressive model factors a sequence law into conditional next-element distributions, while a diffusion model generates through a learned reverse noise process.

Samuel, Zhou and Zou's COLING 2025 study compares five contamination-detection approaches across four language models and eight benchmarks, including a controlled fine-tuning oracle setting.

A code subspace with projector P exactly corrects a specified span of errors E-a precisely when P E-a† E-b P=c-ab P for every error pair.

A stabilizer code is the common fixed subspace of an abelian Pauli subgroup excluding minus identity.

Surface/toric codes measure local stabilizer checks and infer error chains from syndromes.

Fault-tolerant computation must control faults during syndrome extraction, recovery and encoded gates as well as idle storage.

Google Quantum AI's Nature study, online 9 December 2024 and in the 2025 volume, reports a distance-seven surface-code memory using 101 physical qubits with logical error 0.143%±0.003% per cycle.