Foundations

Aeppli and collaborators' archived 2025/2026 report compares 27Al⁺, 171Yb and 87Sr optical clocks with reported total fractional ratio uncertainties at or below 3.2×10⁻¹⁸.

For linear observations z=Aθ+ε with known positive-definite covariance Σ and full-column-rank A, generalized least squares yields θ̂=(AᵀΣ⁻¹A)⁻¹AᵀΣ⁻¹z.

A network model can include per-instrument offsets, temporal drift, shared-reference effects and between-day variability alongside statistical noise.

A discrepancy term δ(x) represents inadequacy of the physical response model, distinct from repeatability noise and uncertain calibration parameters.

Reference-material value assignments have distinct status.

The 12 May 2026 NIST certificate for Great Lakes Fish Tissue certifies total mercury and selected PCB, pesticide and PBDE values on a wet-mass basis.

Detector-based radiation thermometry infers thermodynamic temperature from calibrated spectral radiance, while transfer thermometers and blackbody realizations disseminate a scale.

Nichols' worked compressible mixing-layer comparison specifies two streams, geometry, initial transients and statistical sampling before comparing RANS, LES and hybrid predictions with measurements downstream.

For a normalized trial state of the same Hermitian, lower-bounded Hamiltonian, the Rayleigh energy cannot be below its ground-state energy.

Basis convergence must be tested separately for SCF and correlation contributions.

Small energy and orbital-gradient residuals identify a stationary SCF solution, which may be unstable to allowed orbital rotations.

A larger orbital basis does not remove exchange-correlation functional error.

Counterpoise compares the complex energy with fragment energies evaluated in the complex basis, including partner basis functions without partner nuclei.

Near electronic degeneracy, nuclear motion can couple electronic states through derivative couplings, so a single Born–Oppenheimer potential surface can fail.

A computed vertical electronic excitation is not automatically a measured spectral maximum.

In classical transition-state trajectory counting, recrossing the dividing surface reduces the fraction of nominal crossings that contributes to net reaction; this recrossing transmission coefficient is bounded by one.

In an overdamped diffusive reaction-coordinate model, the flux contains both potential-driven drift and concentration spreading.

The classical harmonic reorganization model gives an electron-transfer barrier depending jointly on driving force and reorganization energy.

Species transport is driven by chemical and electrical contributions to electrochemical potential.

A dilute-solution Nernst–Planck flux combines diffusion, electric migration and convection.

Stefan–Maxwell transport represents friction between different species rather than independent dilute diffusion through an inert solvent.

Current–potential data can reflect charge transfer, bulk migration/diffusion, homogeneous reactions and evolving surface structure.

Molar-mass dispersity is the ratio of mass-average to number-average molar mass.

Size-exclusion results depend on column, solvent and calibration, so retention-based molar masses are not intrinsically absolute.

Branching models locate gelation when continuing connections become self-sustaining.

A Maxwell spring and dashpot in series predicts exponential stress relaxation with a characteristic time.

For linear harmonic loading, the complex modulus separates in-phase elastic storage from quadrature viscous loss.

Corrected diffraction intensities constrain squared structure-factor amplitudes, while phases remain essential for reconstructing electron density.

Phase probabilities and map weighting affect reconstructed density, and model-derived phases can bias the apparent support for that same model.

The Br2+ attosecond study reports coherent vibronic signals and interprets their evolution using electronic/nuclear modeling with anharmonic and spin–orbit contributions.