For constant fractions $f_B,f_D$ in an Einstein-de Sitter universe, $S=f_B{\Delta }_B+f_D{\Delta }_D$ obeys the usual self-gravitating growth equation, while $D={\Delta }_D-{\Delta }_B$ obeys $\ddot{D}+4\dot{D}/(3t)=0$. Thus $S=A{t}^{2/3}+B{t}^{-1}$ and $D=C+E{t}^{-1/3}$. The individual contrasts are ${\Delta }_B=S-f_{D}D$ and ${\Delta }_D=S+f_{B}D$.

A direct band gap compares two energy bands at the same crystal momentum. A global gap instead requires the lowest energy of the upper band to exceed the highest energy of the lower band, allowing the two extrema to occur at different momenta. For example, the periodic toy bands $E_1(k)=A\cos k$ and $E_2(k)=A\cos k+\delta$, with $A>0$ and $0<\delta <2A$, have positive direct separation $\delta$ everywhere but global overlap $2A-\delta$. A positive direct splitting at each Brillouin zone boundary therefore does not alone establish an insulating filled band. At a suitable electron count the overlap can produce both electron and hole pockets.

Consider two overlapping energy bands on a square lattice with spin degeneracy two and primitive reciprocal period $Q$. Let $n$ count the areal electron density in this pair, $n_e$ the occupied density in upper-band electron pockets, and $n_h$ the missing density in lower-band hole pockets. A completely filled lower band provides the reference densityCounting filled states gives $n=n_0-n_h+n_e$, hence $n_e-n_h=n-n_0$. Compensation, meaning equal electron pocket and hole pocket carrier densities, therefore occurs at $n=n_0$. For total pocket areas $A_e,A_h$, the same state counting in a Bloch band gives $n_{e,h}=2A_{e,h}/(2\pi {)}^2$; all equivalent pockets must be counted.

If the geometry is parameterized by $Q=\alpha \sqrt{2\pi n}$ with $\alpha >0$, compensation requires $\alpha =\sqrt{\pi }$. The real-space period is $a=2\pi /Q$, so the filling is $n{a}^2=2$ electrons per primitive unit cell. Compensation can coexist with both kinds of charge carriers; a global band gap is additionally required for a band insulator.

Take the nearest-neighbour hopping $t(a)=-2E_1(1+a/a_1)e^{-a/a_1}$, with $E_1,a_1>0$, in a simple-cubic nearest-neighbour tight-binding band. Substituting into $m^{\ast }=-{\hslash }^2/(2t{a}^2)$ gives the displayed result. With $s=a/a_1$, its logarithmic derivative is $1-2/s-1/(1+s)$, which vanishes only at $s=1+\sqrt{3}>0$. The mass diverges at both positive-domain ends in this formula, so that point is its unique minimum. Its bandwidth has derivative $d\ln W/d\ln a=-s^2/(1+s)$: a small compression increases $W$. The small-$a$ continuation lies beyond reliable localized-orbital tight binding.

For an isotropic valence-band maximum, the electron curvature is negative. The missing-electron excitation has positive charge and positive mass $m_h^{\ast }={\hslash }^2/|E\prime \prime |$. Its cyclotron angular frequency is $eB/m_h^{\ast }$. An anisotropic orbit requires the corresponding cyclotron mass rather than an arbitrary single directional curvature.

Among words with $a$ up-steps and $b$ down-steps, $a>b$, the number whose running balance is strictly positive after every step is $\frac{a-b}{a+b}\left(\genfrac{}{}{0ex}{}{a+b}{b}\right)$. For nonnegative running balance with $a\ge b$, the count is the ballot number $\left(\genfrac{}{}{0ex}{}{a+b}{b}\right)-\left(\genfrac{}{}{0ex}{}{a+b}{b-1}\right)$. These are different boundary conventions; the nonnegative formula follows by reflecting the prefix through the first step below zero.

The first zero of $\cos \kappa L$ makes the homogeneous opposite-end boundary problem singular. Its undepleted-pump treatment admits a nontrivial field without an imposed signal, indicating a parametric oscillation threshold. Depletion, saturation and the actual noise-seeded dynamics determine finite fields near threshold.

For a material parcel of fixed amount inside a much larger thermal and mechanical environment, availability is $A(V)=F(V,T)+p_{\mathrm{bath}}V$. Since $F_V=-p$, its curvature is $A_{VV}=-p_V=1/(V{\kappa }_T)$. The Gaussian fluctuation approximation gives $⟨\delta V^2⟩=k_{B}T{\kappa }_T\bar{V}$. This describes parcel-volume fluctuations, not the geometric volume of a fixed observation box or a globally rigid vessel. A fixed spatial box instead has number fluctuations related to the same compressibility.

A chromium–nickel–molybdenum stainless-steel grade. Annealed and cold-worked conditions can have different irradiation microstructures.

In three-dimensional Euclidean space, contracting an antisymmetric second-rank tensor with the Levi-Civita symbol gives an axial vector. The epsilon contraction identity proves the inverse formula: $\frac{1}{2}{ϵ}_{kij}{ϵ}_{klm}{A}_{lm}=\frac{1}{2}(A_{ij}-A_{ji})=A_{ij}$. With this index convention,Thus $A$ is minus the cross-product matrix for $a\times x$. Proper rotations transform $a$ as a vector; an orientation-reversing orthogonal change adds a determinant sign. Stating the index convention prevents a sign ambiguity when converting an antisymmetric tensor into a cross-product operator.

A thin-film design that suppresses reflected green light more strongly than red and blue can leave a purple residual reflection. A single quarter-wave layer cancels reflections exactly only at its design wavelength and angle under the ideal index-matching condition. reflection colour should be distinguished from the much smaller colour change of the transmitted beam; multilayer designs can broaden the low-reflectance band.

The nonnegative integer $l$ for which a spherical harmonic has squared orbital angular momentum ${\hslash }^{2}l(l+1)$. The angular part of a regular scalar wavefunction has $l=0,1,2,\dots$ and magnetic quantum number $m=-l,\dots ,l$.

An amniote in the lineage containing mammals and their extinct relatives on the mammalian side of the split from sauropsids. The name does not mean a mammalian ancestor was a living reptile.

The reptile-side amniote lineage, including birds. A traditional assemblage excluding birds does not contain all of this ancestry’s descendants.

The corrected Akaike information criterion adds a finite-sample penalty to the Akaike information criterion, where $k$ counts fitted parameters and $n>k+1$. The displayed correction is commonly used for small samples; its exact unbiasedness depends on the model family.

For physical four-momentum $P^{\mu }=md{x}^{\mu }/d\tau$, the covariant-component equation is $d{P}_{\mu }/d\tau =({\partial }_{\mu }{g}_{\nu \sigma })P^{\nu }{P}^{\sigma }/(2m)$. The equation without $1/m$ instead uses a tangent parametrization, or the affine parameter $\lambda =\tau /m$ when the tangent is the physical momentum.

A change $\lambda \mapsto A\lambda +B$ rescales a geodesic tangent and all its linear Killing constants by $1/A$. A null norm remains zero and cannot fix $A$, unlike proper-time normalization of a material particle. Ratios of the constants, such as the photon impact parameter $b=|h|/(ck)$ in Schwarzschild spacetime, are independent of the affine scale. A physically normalized four-momentum fixes the photon energy and hence selects a tangent scale.

An affine parameter on a timelike geodesic can be scaled to proper time. With metric signature $(+---)$ this fixes the tangent-vector norm to $c^2$. A further constant rescaling preserves the affine geodesic equation but changes that norm; conserved energy and angular-momentum constants must be rescaled consistently. The unit term in a proper-time-normalized radial energy equation is consequently a normalization choice, not a property of every affine parameter.

If a hole gains mass $\Delta M_{\mathrm{BH}}$ at constant radiative efficiency $\eta =L/({\dot{M}}_{\mathrm{in}}{c}^2)$, the radiated energy is $\eta \Delta M_{\mathrm{BH}}{c}^2/(1-\eta )$. For negligible seed mass, final hole mass $\alpha M$, and total three-dimensional galaxy dispersion $\sigma$, its ratio to $|W|\simeq M{\sigma }^2$ is $\eta \alpha (c/\sigma {)}^2/(1-\eta )$. The net virial unbinding energy is half $|W|$ when surface terms are negligible.

For an approximately pressure-balanced cavity containing relativistic fluid with $\gamma =4/3$, internal energy plus displacement work is $4pV$. Dividing by an estimated inflation or buoyancy age gives a mechanical-power diagnostic. Projection, pressure gradients, the equation of state, age estimates and additional shock energy affect the inference; cavity power is not identical to locally dissipated heat.