For a perturbation phase $m\varphi +\int kdr-\omega t$, a constant-phase pattern rotates at ${\Omega }_p=\omega /m$. At corotation the local orbit angular speed equals this value. One observed galaxy can contain several patterns with distinct angular speeds and lifetimes.

A self-gravitating disk can transiently amplify a leading disturbance as differential rotation shears it into a trailing disturbance. Its strength depends on disk self-gravity, random motions and disturbance wavelength. Amplification does not by itself establish one indefinitely steady spiral.

Two radii with different orbital frequencies develop a relative angular displacement $\Delta \varphi =\Delta \Omega t$. A spiral made of a fixed set of stars therefore winds increasingly tightly. A density pattern need not follow that material winding because matter can pass through it.

A stellar disk does not respond to a density wave as a collisional gas with a local pressure law. Its stars follow individual orbits, and their spread in radial motions partly averages away the gravitational response to short-wavelength perturbations. The appropriate calculation uses a perturbed distribution function rather than only a fluid sound speed.

A characteristic parameter is $\chi =(k{\sigma }_R/\kappa {)}^2$, comparing the radial epicycle scale with the perturbation wavelength. Orbital averaging reduces the response to perturbations whose wavelength is small relative to the stellar excursions. Merely replacing the fluid sound speed with stellar velocity dispersion misses that orbital averaging and can give the wrong stability threshold.^[ref]

A protein assembly that inhibits productive anaphase-promoting complex–Cdc20 activity while kinetochore-associated checkpoint signalling remains active.

A checkpoint component participating in inhibition of Cdc20 as part of the mitotic checkpoint complex.

Changes in its interactions support assembly of a signal that restrains anaphase-promoting activity when chromosome attachment is inadequate. This converts an unsatisfied local attachment state into a delay of the global chromosome-separation decision. ^[ref]

A checkpoint regulator involved in attachment surveillance and inhibitory complexes controlling anaphase-promoting complex–Cdc20.

Checkpoint signalling restrains activation of the anaphase-promoting complex until suitable attachment conditions are reached. Inhibitory-complex assembly and the physical correction of attachments are related but distinct tasks, so one checkpoint readout need not establish normal chromosome alignment. ^[ref]

A checkpoint component participating in kinetochore-associated signalling and mitotic checkpoint complexes.

Its binding interactions help localize checkpoint machinery and assemble inhibitory signals that delay anaphase when attachment is inadequate. A checkpoint component is therefore part of a communication system between chromosome attachment state and cell-cycle progression. ^[ref]

For the auxiliary-field convention with cost $M^2/(2U)$ and coupling $-M\bar{\psi }{\sigma }_z\psi$, the homogeneous saddle point equation equates M to U times spin polarization per site. M has energy units. It is not itself a magnetic moment, and its coefficient changes if the contact interaction or density-of-states spin convention is changed. The exact three-component decoupling must be distinguished from selecting one constant spin direction as a mean-field approximation.

For perfect transmission above threshold $E_n$, integrating $-df/dE$ gives $G_n=(e^2/h)f(E_n)$. Thermal occupation rounds a zero-temperature conductance step over an energy range of order $k_{B}T$.

The distance over which a nonequilibrium spin imbalance decays appreciably in a diffusive conductor. In a simple diffusion-relaxation model it is ${\ell }_s=\sqrt{D{\tau }_s}$; interfaces can contribute additional spin-memory loss.

Loss of transported spin polarization at an interface, distinct from bulk spin relaxation in an adjacent layer.

A transverse magnetic field rotates a nonequilibrium spin polarization while diffusion and spin relaxation redistribute it. The field dependence of a detector signal can constrain spin lifetime and transport parameters. A distribution of diffusion times and rotation of the injecting/detecting electrodes must be included when relevant.

For spin one-half, the Pauli representation makes every diagonal quadratic component a multiple of the identity. Thus $A{S}_z^2+B(S_x^2-S_y^2)=A{\hslash }^{2}I/4$ cannot split the doublet or cause spin reversal. This conclusion is special to this quadratic Hamiltonian and does not exclude linear Zeeman dynamics.

For uncoupled centres with approximately equal $g=2$, squared magnetic moments add: ${\mu }_{\mathrm{eff}}/{\mu }_B=2\sqrt{{\sum }_i{S}_i(S_i+1)}$. This does not assume that the individual spins add to a definite total spin; appreciable exchange interaction invalidates the independent-centre approximation.

A spin gap is a nonzero minimum energy required to reach a specified magnetic excitation above the ground state. One must specify which spin sector or measurement couples to that excitation; not every excitation gap is a spin gap.

Two antiferromagnetically coupled spin-$1/2$ particles illustrate it. For $H=J{\mathbf{S}}_1\cdot {\mathbf{S}}_2$ with dimensionless spin operators and $J>0$, the singlet has energy $-3J/4$ and the triplet has $J/4$, giving gap $J$. Thermal magnetic response is suppressed when $k_{B}T$ is well below that scale. Extended systems require their collective excitation spectrum rather than this isolated-pair argument alone.

A mixture whose orthohydrogen and parahydrogen mole fractions are fixed on the observation timescale, although each species can equilibrate over its own rotational levels. Its molar heat capacity is $x_o{C}_o+x_p{C}_p$. normal hydrogen retained during cooling has $x_o=3/4$ and $x_p=1/4$. A mixture initially prepared at some other equilibrium composition can also be frozen; its subsequent heat response does not include a changing-composition term.

Interconversion of orthohydrogen and parahydrogen, which changes the coupled nuclear-spin and rotational populations. An active conversion mechanism allows the mole fractions to follow equilibrium as temperature changes; slow conversion instead gives a frozen ortho-para mixture. Whether conversion remains active during a thermal measurement must therefore be distinguished from how the initial sample was prepared.

For positive membrane tension, the bound-cap energy is convex in indentation. Its constrained minimum is the unconstrained value $(\bar{w}-2)/\bar{\sigma }$ clipped to $[0,2]$. No wrapping occurs at $\bar{w}\le 2$, partial wrapping at $2<\bar{w}<2+2\bar{\sigma }$, and full wrapping at $\bar{w}\ge 2+2\bar{\sigma }$. At zero tension the expression is linear, with all indentations degenerate at $\bar{w}=2$. These phase boundaries belong to the specified cap model and can move when free-membrane and neck energies are included.

An initially expanding overdense region reaches a maximum physical radius before contracting. In a spherical pressureless model its radial expansion velocity vanishes there. The background universe can continue expanding throughout the local turnaround and subsequent collapse.