The reduction in size across the lanthanide series associated with incomplete screening by $4f$ electrons; it contributes to size similarities and contractions in subsequent $5d$ elements.

A periodic trend is a systematic variation in an atomic or chemical property across the periodic table. Examples include atomic size, ionization energy, and electronegativity.

Electron-shell filling, screening, and nuclear charge help explain broad patterns, but definitions and exceptions matter. An atomic radius inferred from a covalent bond is not the same measurement as an ionic radius in a crystal, so trends should compare like quantities.

A lanthanide is an element in the series from lanthanum, atomic number 57, through lutetium, atomic number 71. The progression is associated with filling the 4f shell and the systematic size change called the lanthanide contraction.

Many lanthanides favour trivalent ions with similar coordination chemistry. Their similarity makes separation demanding, while differences in ionic size and accessible oxidation states permit selectivity.

An actinide is an element in the series from actinium, atomic number 89, through lawrencium, atomic number 103. The series is associated with occupation of 5f-derived electronic states, although individual ground-state configurations vary.

Variable oxidation states and ionic sizes affect coordination and separation chemistry. Radioactive isotopes add a distinct nuclear aspect to applications involving nuclear fuel and waste materials.

An integer equal to the third power of another integer, such as $8=2^3$ or $-27=(-3{)}^3$.

Experience considered from the experiencing subject's standpoint. Observable performance does not uniquely specify its qualitative character.

A dependable relation between an action and its sensory consequences. A mirror generates tightly coupled visual changes that can support learning about the relation between body movement and reflected movement.

A bacterial cell-envelope assembly involved in oriented lateral peptidoglycan synthesis in many rod-shaped species.

Coordinating synthesis around the cell sidewall helps maintain an elongated geometry. This differs from the divisome, which constructs the new boundary at a division site. Defects in elongation and division can therefore produce different changes in cell shape. ^[ref]

A plastid RNA-binding protein that can protect a bound region against exonucleolytic decay from either direction, helping define stable transcript termini.^[ref]

For $H=\mathbf{d}\cdot \mathbf{\sigma }$ with $\mathbf{d}\ne 0$, the eigenvalues are $\pm |\mathbf{d}|$ and the projectors are $P_s=(I+s\hat{\mathbf{d}}\cdot \mathbf{\sigma })/2$. A normalized eigenstate has $⟨\mathbf{\sigma }⟩=s\hat{\mathbf{d}}$. At $\mathbf{d}=0$ the spectrum is degenerate and that direction is undefined.

For diffusion constant $D>0$, Gaussian increments have variance $2D\Delta t$. Their convolution gives a path weight proportional to $\exp [-\sum (\Delta x{)}^2/(4D\Delta t)]$ with measure factor $(4\pi D\Delta t{)}^{-M/2}$ for $M$ slices. The continuum kinetic action is notation for this limiting measure; typical Brownian paths have no ordinary derivative.

In a bosonic coherent-state path integral, coherent-state resolutions between imaginary-time evolution factors turn a normal-ordered grand Hamiltonian $K=H-\mu N$ into its complex symbol. Coherent-state overlaps supply the first-order temporal term. A thermal trace makes the fields periodic in imaginary time. The chemical potential couples to number, and interactions retain their normal-ordering convention.

For a one-electron P-to-S transition with nuclear spin zero, ten electric-dipole sublevel channels have finite strength at weak nonzero field. Six retain finite strength in the complete orbital-spin decoupling limit. Channel count is distinct from spectral frequency count: retaining first-order residual spin-orbit interaction energy gives five distinct frequencies, with two central channels degenerate.