The pair count of a two-mode squeezed vacuum has $P(n)=(1-r)r^n$, where $r=|q{|}^2<1$. Its mean is $r/(1-r)$ and its variance is $r/(1-r{)}^2$. Equal mode occupations follow from pair creation, not from independent particle and antiparticle draws.

Suppose a model replaces a set of free-electron boundary energies $E^{(0)}$ by lower and upper edges $E^{(0)}-V$ and $E^{(0)}+V$, with $V>0$. If these are the relevant global band edges, the indirect band gap is $(E_{\min }^{(0)}+V)-(E_{\max }^{(0)}-V)$, giving the displayed expression. A positive direct gap $2V$ at each crossing need not be enough: a band insulator needs $2V>E_{\max }^{(0)}-E_{\min }^{(0)}$. This is a band-edge estimate, not a general three-dimensional nearly-free-electron theorem; multiple Bragg planes at a corner can require a larger coupling matrix.

An operation fidelity used to compare a physically implemented quantum gate with its intended operation. A quantitative value requires a declared input-state average or another explicit channel-comparison convention; different conventions need not give the same number.

For rigid spin-degenerate channel transmissions and ${\mu }_{S,D}=E_F\pm eV/2$, differentiating $I=(2e/h){\sum }_n{\int }_{{\mu }_D}^{{\mu }_S}{T}_n(E)dE$ gives $G_d=(e^2/h){\sum }_n[T_n({\mu }_S)+T_n({\mu }_D)]$. If the two edges sample neighboring integer channel counts, $G_d$ is a half-integer multiple of $2e^2/h$. Explicit bias dependence of the potential adds terms; the apparent fraction does not imply fractional carriers.

A channel supporting ballistic electron transport carries incident populations set by different source and drain chemical potentials, rather than an everywhere-equilibrated local distribution with an Ohmic voltage gradient. Its electrostatic barrier responds self-consistently to contacts, gates and electrostatic screening. A symmetric rigid-barrier model places reservoir energies at $E_F\pm eV/2$ relative to the barrier. Asymmetric partition or bias-dependent transmission changes the weights in the differential electrical conductance, and must be included when extracting an energy scale.

In a rigid spectrum, source and drain threshold lines in a bias-versus-gate map satisfy $E_n(V_g)={\mu }_S$ or ${\mu }_D$. Where adjacent thresholds align with opposite edges of the bias window, their spacing equals the window width $e|V|$. For symmetric bias and spin degeneracy, neighboring channel counts at those edges give a differential electrical conductance halfway between integer $2e^2/h$ plateaus. The dc ratio $I/V$ is a different observable; barrier motion and heating can modify the interpretation.

An equatorial disclination loop surrounding a roughly spherical inclusion with homeotropic anchoring. A meridional section cuts the loop twice. It can match a radial near-surface director to a uniform far field.

A plot of successive vibrational spacings against a vibrational index. Extrapolation toward zero spacing estimates a dissociation limit under an assumed spacing law. The integration bounds and whether the energy is measured from the potential minimum or vibrational ground state must be specified.

A periodic chain of $N$ sites and length $Na$ has allowed signed wave numbers separated by $2\pi /(Na)$. The continuum state density on the signed axis is $Na/(2\pi )$. Combining opposite propagation directions onto $0<|q|<\pi /a$ doubles it to $Na/\pi$. This counts the same $N$ modes, not two independent acoustic phonon branches.

A swept gas shell of fraction $f$ at radius $R$ in a galaxy of one-dimensional dispersion $\sigma$ has mass $2f{\sigma }^{2}R/G$. Interior non-gas gravity supplies weight $4f(1-f){\sigma }^4/G$. Fully intercepted single-scattering Eddington momentum overcomes it when $M_{\mathrm{BH}}>f(1-f){\sigma }^4{\sigma }_T/(\pi m_p{G}^2)$. This neglects shell self-gravity, central-hole gravity and stored outward kinetic energy; weaker coupling raises the required luminosity.

A calcium zirconium titanate with ideal formula $\mathrm{CaZrT}{\mathrm{i}}_2{\mathrm{O}}_7$. Its structural variants and substitution mechanisms are important in actinide-bearing ceramic waste forms.

A uranium titanate with ideal composition $\mathrm{UT}{\mathrm{i}}_2{\mathrm{O}}_6$. It occurs in natural actinide-bearing minerals and can appear in titanate ceramic waste forms.

A mineral whose structure contains silicon–oxygen units. Different linkages give isolated groups, chains, sheets or extended frameworks. Refractory silicate grains are an important solid component of a protoplanetary disk.

The Triassic is the earliest Mesozoic period, approximately $251.9$ to $201.4$ million years ago.^[ref] It precedes the Jurassic. In a local sedimentary rock sequence, a missing part of this interval may reflect nondeposition or later erosion; the absence of a named interval in one section does not imply that no time elapsed there.

The Jurassic is the Mesozoic period between the Triassic and Cretaceous, approximately $201.4$ to $143.1$ million years ago.^[ref] Its position in a rock sequence can be established by correlated stratigraphic evidence even where no directly datable volcanic layer occurs. Numerical calibration and recognition of the interval in local strata are therefore distinct tasks.

The Cretaceous is the youngest Mesozoic period, approximately $143.1$ to $66$ million years ago in the dated chart used here.^[ref] A numerical boundary estimate is a calibration of the geological time scale, not an assertion that every local geological process changed simultaneously. Stratigraphy establishes correlations; radiometric dating supplies numerical age constraints.

The equal-area condition is naturally $\int (p-p_{\mathrm{coex}})dv=0$ in volume per particle $v$. In number density $n=1/v$, it becomes ${\int }_{n_g}^{n_l}[p(n)-p_{\mathrm{coex}}]dn/n^2=0$. Unweighted area in a pressure-versus-density plot is not the Maxwell construction.

For $-y^{\prime \prime }=\lambda x^{-2}y$ on $1<x<e^{\ell }$ with zero endpoint values, the natural weighted inner product has measure $dx/x^2$. Set $v=\ln x$ and $y(x)=\sqrt{x}Z(v)$. The chain rule gives $x^2{y}^{\prime \prime }=\sqrt{x}(Z^{\prime \prime }-Z/4)$, so $-Z^{\prime \prime }+Z/4=\lambda Z$ with zero endpoints on $(0,\ell )$. Moreover $|y{|}^{2}dx/x^2=|Z{|}^{2}dv$, so the transformation preserves the norm. The ordinary sine orthonormal eigenbasis therefore gives exactly the displayed spectrum and eigenfunctions for $n\ge 1$. The factor $\sqrt{x}$ and the additive $1/4$ distinguish this problem from the operator $-x^2{y}^{\prime \prime }-x{y}^{\prime }$ with measure $dx/x$.

This instance of Rodrigues' formula constructs the Legendre polynomial for every nonnegative integer $l$. Put $v=(x^2-1{)}^l$; then $(x^2-1)v^{\prime }=2lxv$. Differentiating $l+1$ times by the Leibniz rule gives $(x^2-1)y^{\prime \prime }+2x{y}^{\prime }-l(l+1)y=0$ for $y=v^{(l)}$, which is the Legendre differential equation after reversing signs. The leading term of the normalized derivative is $(2l)!x^l/[2^l(l!{)}^2]$, proving its polynomial degree is exactly $l$. At $x=1$, the product $(x+1{)}^l(x-1{)}^l$ contributes only the term with all $l$ derivatives on the second factor, giving $v^{(l)}(1)=2^{l}l!$. Thus the normalization is $P_l(1)=1$.