Conversion between differentiated cellular identities. Whether a progenitor-like intermediate is traversed is an experimental question rather than part of a universal mechanism.

Changes in intestinal epithelial potential that allow previously committed cells to restore a renewing stem-cell compartment under particular conditions.

For total source electrons $S$, $n$ aperture pixels, sky-plus-dark rate $b+d$, time $t$, $M$ reads with rms $r$, and an independent mean-background estimate of variance ${\sigma }_{\hat{B}}^2$ per pixel, the aperture variance is $S+n(b+d)t+nM{r}^2+n^2{\sigma }_{\hat{B}}^2$ before additional calibration terms. The last factor is $n^2$ because one uncertain background mean is subtracted from all aperture pixels. Pixel correlations require covariance terms.

For the sequential decay defining the cascade-decay invariant-mass endpoint, momentum conservation gives the displayed eventwise identity. Thus $m_{At}^2$ has minimum $m_A^2{m}_C^2/m_B^2$ and maximum $m_A^2+m_C^2-m_B^2$, attained at the opposite relative angles from the $s,t$ pair endpoints. The final A particle is massive, so its lower pair-mass bound need not vanish.

The reversible heat-engine efficiency between absolute temperatures $T_H$ and $T_C$, ${\eta }_C=1-T_C/T_H$. Real conversion between the same reservoirs cannot exceed it.

The angle of the wrapping vector from the zigzag direction of graphene. For chiral indices $0\le m\le n$, $\theta ={\tan }^{-1}[\sqrt{3}m/(2n+m)]$, ranging from zero for zigzag tubes to 30 degrees for armchair tubes. Together with diameter, it determines the discrete wrapping geometry, but does not alone determine the observed colour of a nanotube suspension.

At fixed particle number and volume, a classical monatomic ideal gas has $U=3N{k}_{B}T/2$ and $C_V=3N{k}_B/2$. The canonical energy fluctuation identity $\mathrm{Var}(E)=k_B{T}^2{C}_V$ therefore gives ${\sigma }_U/U=\sqrt{2/(3N)}$. Absolute fluctuations grow as $\sqrt{N}$, while relative fluctuations decrease as $N^{-1/2}$. A particle reservoir changes the fluctuation problem because energy and particle-number fluctuations are then correlated.

For unpolarized incident particles, the resonant cross section sums over the resonance's magnetic substates and averages over the incident spin states. This gives the displayed statistical weight under the conventional Breit-Wigner resonance cross section normalization. A spin-zero resonance formed from two spin-half particles has $g=1/4$, whereas a spin-one resonance has $g=3/4$. This counting is distinct from final-state color charge multiplicities.

A direct lattice vector is an integer linear combination of primitive translation vectors ${\mathbf{a}}_i$ of a Bravais lattice. Translating by such a vector preserves the lattice. Its dot product with every reciprocal lattice vector is an integer multiple of $2\pi$ in the angular-wavevector convention.

The set $\{n_1{\mathbf{a}}_1+n_2{\mathbf{a}}_2:n_1,n_2\in \mathbb{Z}\}$ for two independent planar translation vectors. Its primitive-cell area is $|\det ({\mathbf{a}}_1,{\mathbf{a}}_2)|$.

A tetragonal Bravais lattice represented by conventional-cell points $(0,0,0)$ and $(1/2,1/2,1/2)$, with $a=b\ne c$ and right angles. A motif attached to each point determines the atomic structure and can give more than two atoms per conventional cell.

A magnetic field can close the lowest gap of a dilute bosonic triplet branch. In a shifted-parabola model with actual chemical potential zero, ${ϵ}_k={\hslash }^2{k}^2/(2m)+\nu n-\eta (H-H_g)$ and repulsion $\nu >0$ prevent indefinite density growth. The normal minimum reaches zero when $\eta (H_c-H_g)=\nu n$. At that point the finite thermal excited population saturates and macroscopic ground-state occupation can develop.

With the ground-state energy set to zero, the chemical potential of an ideal Bose gas cannot exceed zero. The limiting population of excited states as it approaches zero is the saturation particle number, provided the integral converges. Additional particles occupy the ground state macroscopically. For a three-dimensional uniform box, $N_c=\zeta (3/2)V/{\lambda }^3$ with thermal wavelength $\lambda =h/\sqrt{2\pi m{k}_{B}T}$.

For a partially condensed equilibrium ideal Bose gas in a fixed three-dimensional box, the energy scales as $T^{5/2}$. Unbiased random particle removal keeps mean energy per particle fixed, so re-equilibration gives $T\propto N^{2/5}$ until condensation disappears. Removal of an excited particle can be followed by partial replenishment from the zero-energy condensate. This differs from selective evaporative cooling of atoms, which preferentially removes high-energy atoms and reduces mean energy per particle.

For a one-component ideal Bose gas with masses $(m,m,\gamma m)$, insert the anisotropic ideal-gas density of states in the excited-state number integral at chemical potential zero. The substitution $x=\epsilon /(k_{B}T)$ gives $n=\sqrt{\gamma }(m{k}_{B}T/(2\pi {\hslash }^2){)}^{3/2}\zeta (3/2)$, which yields the displayed transition temperature. Thus at fixed density and the two lighter masses, $T_c\propto {\gamma }^{-1/3}$. This continuum ideal-gas result does not supply a finite-temperature transition for decoupled two-dimensional layers.

Compare uniform three-dimensional ideal gases at the same number density n, with fermion mass $m_F$, spin degeneracy $g_F$, boson mass $m_B$ and a single boson spin component. The Fermi temperature has $k_B{T}_F={\hslash }^2(6{\pi }^{2}n/g_F{)}^{2/3}/(2m_F)$, while Bose saturation gives $k_B{T}_B=2\pi {\hslash }^2[n/\zeta (3/2){]}^{2/3}/m_B$. Eliminating density yieldsThe degeneracy is essential; using a spinless Fermi sea changes the numerical estimate. This is an ideal-gas comparison and does not remove interaction effects in a dense liquid.

With planet-to-star mass ratio $q$ and disk aspect ratio $h$, $r_B/a=q/h^2$ and $r_H/a=(q/3{)}^{1/3}$. Equating the two gives $q=h^3/\sqrt{3}$. The thermally limited radius grows faster with planet mass than the tidally limited radius; an envelope outer scale is often the smaller of the two.

A limiting heat flux associated with a transition to a less effective boiling heat-transfer regime under specified conditions. Its value depends on fluid properties, pressure, flow, heating and geometry; a single number cannot be transferred indiscriminately between different flow passages.

A strictly increasing sequence $s_i$ for which $s(X_t)$ is a martingale along the birth-death chain before hitting its boundary. With upward and downward probabilities $p_i,q_i$, harmonicity means $p_i(s_{i+1}-s_i)=q_i(s_i-s_{i-1})$. This coordinate is useful for hitting probabilities and optimal stopping.

Birefringence produced by subwavelength structure rather than solely by intrinsic crystalline anisotropy. For ideal dielectric lamellae, continuity of tangential electric field gives ${ϵ}_{\parallel }=f{ϵ}_1+(1-f){ϵ}_2$, while continuity of normal electric displacement field gives ${ϵ}_{\perp }^{-1}=f/{ϵ}_1+(1-f)/{ϵ}_2$. The unequal effective permittivities give different principal refractive indices.