The minimum local departure from nucleate boiling ratio over the specified core region or assessment. It depends on local heat loading and coolant conditions, not only on average core power.

A member of the lineage including mammals and close extinct relatives such as Morganucodon. Usage must distinguish this broader grouping from a living mammalian crown group.

A cynodont clade containing mammaliaforms and some close extinct relatives. It is broader than the living mammalian crown group.

In a model of carriers sampling alternating magnetic and nonmagnetic layers, assume spin conservation, comparable carrier transport weights and negligible extra interface scattering. Averaging relaxation rates by the time spent in each material then gives a channel electrical resistivity $(t_1{\rho }_{\mathrm{FM},\mathrm{s}}+t_2{\rho }_{\mathrm{NM}})/(t_1+t_2)$. Here all constituent resistivities are per spin channel. A long mean free path compared with a period motivates interlayer sampling; it does not by itself prove the other assumptions. This is a homogenized current-in-plane model, rather than an independent parallel-layer limit or a current-perpendicular-to-plane resistor chain.

When attachment is much slower than exterior diffusion, the surface concentration stays close to the reservoir value. An attachment coefficient $k_s$ with units of velocity gives nearly constant radial growth in the dilute-feed approximation. For a sphere the ratio $k_{s}a/D$ distinguishes attachment control from transport control. A finite attachment rate leads to a Robin boundary condition rather than a fixed equilibrium surface concentration.

A small noncondensed fraction provides a useful global expansion parameter for a weakly interacting Bose gas, but does not bound every long-wavelength mode individually. In Bogoliubov theory, mode amplitudes can grow as momentum approaches zero, while their integrated contribution remains finite in some dimensions and regimes.

A fluctuation integral includes both the small-momentum behavior and the momentum-space measure. For example, an integrand proportional to $1/k$ contributes ${\int }_{0}dk{k}^{d-2}$ in $d$ spatial dimensions, which converges at zero only for $d>1$. Thermal factors can make the infrared behavior more singular. Thus condensate fluctuation power counting must be supplemented by checking the relevant integrals, system size and temperature. The uniform phase mode and fixed particle number constraint also require their own treatment.

Molar crystal volume follows from $V_m=N_A{V}_{\mathrm{cell}}/Z$ with $Z$ atoms per cell for an elemental solid. Combining two such values with their molar entropy difference gives $\mathrm{d}P/\mathrm{d}T=\Delta S_m/\Delta V_m$. Ambient values supply only an approximation to a high-pressure coexistence slope, not a phase-boundary intercept.

Floating ice with submerged volume fraction $f$ has density $f{\rho }_l$ by Archimedes' principle. Its volume change on melting is therefore $v_l-v_s=(1-1/f)/{\rho }_l<0$. An enthalpy cycle gives $L_f=L_s-L_v$ for quantities consistently expressed per mass at the same temperature. Treating these latent heats and specific volumes as constant and integrating the Clausius-Clapeyron relation gives the displayed pressure estimate. Using a vaporization latent heat extrapolated from a different temperature carries that approximation into the inferred fusion heat.

A cylindrical model nucleus of diameter $d$ and height $h$. Its free energy is $\pi d^{2}h\Delta g/4+\pi d^2{\gamma }_f/2+\pi dh{\gamma }_e$, where $\Delta g$ is the signed bulk free-energy change per volume.

The maximum free-energy cost along a nucleation pathway, at its critical nucleus. A pathway with fixed shape is a constraint and need not coincide with an unconstrained saddle point.

In the spherical capillarity approximation, the nonzero maximum of $\Delta G=4\pi r^2\gamma +(4\pi /3)r^3\Delta g_V$ lies at $r^{\ast }=-2\gamma /\Delta g_V$ when $\Delta g_V<0$. Smaller nuclei tend to shrink and larger ones tend to grow. Surface anisotropy and molecular-scale structure limit the approximation.

The identities $K_{+}{K}_{-}=N_a{N}_b$ and $K_{-}{K}_{+}=(N_a+1)(N_b+1)$ give the displayed quadratic Casimir operator. It commutes with all generators because the number difference does. A fixed number-difference sector therefore has a fixed Casimir value even though pair creation changes its total occupation.

For angular Fourier modes of integer order $n\ge 2$, the biharmonic equation in polar coordinates has radial basis $r^n,r^{-n},r^{n+2},r^{2-n}$. Apply the Laplacian twice to obtain the displayed indicial equation. For $n=0$, repeated roots give the radial basis $1,\log r,r^2,r^2\log r$; for $n=1$, the basis is $r^{-1},r,r\log r,r^3$. These are Cauchy-Euler equations. Regularity at the origin and boundary conditions select the admissible powers. For $n=2$, a constant radial factor still multiplies $\cos 2\theta$ or $\sin 2\theta$, so it is bounded but fails continuity at the origin.

A member of the living crocodilian lineage, including crocodiles and their closest living relatives. birds are its living sister lineage within Archosauria.

A member of the dinosaur lineage, including birds. Restricting the word to extinct nonavian forms excludes descendants.

The avian total group: Aves and all extinct archosaurs more closely related to them than to living crocodilians. Under standard relationships this includes pterosaurs and nonavian dinosaurs.

A pterosaur is a member of an extinct group of flying archosaurs. Its wing membrane was supported chiefly by an elongated fourth finger. That construction differs from the feathered wing of a bird and the multi-digit membrane support of a bat.

For equal ring halves with potentials $V_0$ and zero, the zero-current Thomas–Fermi bulk solution has $\mu ={\mu }_0+V_0/2$ and populations $N_{L,R}=N(1\mp V_0/2{\mu }_0)/2$ while both halves are occupied. For $V_0>2{\mu }_0$ the raised half is depleted and $\mu =2{\mu }_0$ in this approximation.

For $S=⟨3\mathbf{s}{\mathbf{s}}^{\mathsf{T}}-I⟩=Q(3\mathbf{n}{\mathbf{n}}^{\mathsf{T}}-I)$ and unit molecular orientations $\mathbf{s}$, contraction with $\mathbf{n}$ gives $2Q=3⟨(\mathbf{s}\cdot \mathbf{n}{)}^2⟩-1$. Since the squared projection is between zero and one, the displayed bounds follow. The eigenvalues in this normalization are $2Q,-Q,-Q$. A differently normalized nematic order parameter has correspondingly rescaled bounds; its numerical coefficients cannot be transferred unchanged.