The integral of the receiver operating characteristic's true positive rate with respect to its false positive rate. Equivalently, the probability that a randomly selected positive is ranked above a randomly selected negative, with half weight for ties. It measures discrimination rather than probability calibration or utility at a particular threshold.

The coherent coupling frequency defined by the chosen driven two-level system Hamiltonian. With $H=\hslash (-\Delta {\sigma }_z+\Omega {\sigma }_x)/2$, resonance gives population oscillations proportional to ${\sin }^2(\Omega t/2)$. Detuning replaces the oscillation frequency by $\sqrt{{\Omega }^2+{\Delta }^2}$ and reduces its contrast. Stating the Hamiltonian avoids factor-of-two ambiguities in the definition of $\Omega$.

For positive gyromagnetic ratio in static field $B\hat{z}$, transverse components $B_1\cos (\omega t+\phi )$ and $-B_1\sin (\omega t+\phi )$ rotate with the spin precession. Their rotating-frame Hamiltonian operator is time independent, with transverse Rabi angular frequency $\Omega =\gamma B_1$. A linearly polarized field of peak amplitude $B_{\mathrm{peak}}$ contains two oppositely rotating components and gives resonant $\Omega =|\gamma |B_{\mathrm{peak}}/2$ in the rotating-wave approximation. Negative gyromagnetic ratio requires opposite rotation sense.

A circularly driven spin becomes time-independent after a rotation $R=e^{-i\omega t{S}_z/\hslash }$. The transformed Hamiltonian operator is $R^{†}HR-i\hslash R^{†}\dot{R}$, so the longitudinal frequency is shifted by $-\omega$. This is an exact change of frame for a circular drive. The rotating-wave approximation is a separate approximation needed when a drive also contains a counterrotating component.

Suppose the unknown ideal elements are a linear polarizer and a quarter-wave plate, and the two unknown sources are unpolarized and circularly polarized. Test both element orders while rotating their relative axes and monitoring output intensity. Only circular input followed by the quarter-wave plate and then the polarizer gives complete modulation: the plate makes linear light and the analyzer obeys Malus's law. Polarizer-first output is constant half-intensity because the following lossless plate changes polarization but not intensity. Unpolarized input is constant in either order. This identifies both elements and both sources without comparing source brightnesses.

The ratio ${\nu }_{\mathrm{fill}}=n_{s}h/(eB)$ for sheet carrier density $n_s$. Each resolved spin branch has degeneracy $eB/h$ per unit area; an unresolved orbital level can contain two spin branches. A plateau index need not equal total filling everywhere on a finite-width plateau.

If a condensate order parameter stays nonzero along a closed loop, its phase changes by an integer multiple of $2\pi$. Broken rotational symmetry does not destroy this single-valuedness constraint. In an inhomogeneous ring the winding integer need not label an angular-momentum eigenstate.

For $S=A\cos (q\theta )$ with independent standard uncertainties, linear propagation gives ${\sigma }_S^2={\cos }^2(q\theta ){\sigma }_A^2+[qA\sin (q\theta ){]}^2{\sigma }_{\theta }^2$. Angular uncertainty must be in radians when using this derivative. Covariance adds cross terms for correlated inputs. Strict error bounds instead add absolute first-order contributions; they do not justify Gaussian significance tests.

For axial moment of inertia $I=m(a^2+b^2)/2$, linear sensitivity to mass and outer/inner radius is $dI=(a^2+b^2)dm/2+mada+mbdb$. Independent standard uncertainties give ${\sigma }_I^2=[(a^2+b^2){\sigma }_m/2{]}^2+(ma{\sigma }_a{)}^2+(mb{\sigma }_b{)}^2$. Correlated measurements instead require the full covariance quadratic form. For deterministic bounds, add the absolute linear contributions rather than their squares.

For air approximated by index one, $n=\sin i/\sin r$, so $dn/n=\cot idi-\cot rdr$. Taking the variance gives the displayed first-order error formula, with angles and their errors in radians. Independence removes the covariance term; hard error bounds require a different combination rule. Common angular-zero errors can produce covariance and should not automatically be added in quadrature.

For $z=\sqrt{R^2+X^2}$ with reactance $X=\omega L$, linear differentiation gives $dz=(RdR+XdX)/z$. Independent standard uncertainties give the title formula; correlated inputs add $2RX\mathrm{Cov}(R,X)/z^2$. Hard error bounds instead add absolute linear contributions. The meaning of an error bar is an additional measurement assumption, not a consequence of the impedance formula.

The fraction of predicted positives that are genuinely positive. It depends on prevalence as well as the true positive rate and false positive rate.

The fraction of genuinely positive cases recovered by a classifier. The denominator counts all reference-positive cases, $TP+FN$, rather than positive predictions. In a diagnostic setting this is diagnostic sensitivity.

A plot of classification precision against classification recall as a threshold changes. An uninformative ranking has expected precision equal to the positive prevalence. Comparisons should retain the same evaluation population or explicitly account for changed prevalence.

For the bounded rank-one operator $Kx=a\phi (x)$, insert the powers of a rank-one operator into the convergent operator exponential power series. Writing $c=\phi (a)$ gives the displayed formula for $c\ne 0$. At $c=0$ the quotient has limit $t$, so $e^{tK}=I+tK$. This includes the zero operator and the nonzero nilpotent operator cases without dividing by zero.

Conversion of relatively unreactive chlorine reservoirs into forms that produce active radicals in sunlight. Heterogeneous reactions on appropriate particle surfaces can turn hydrogen chloride and chlorine nitrate into photolabile species. Available surface composition, temperature, reservoir supply and subsequent illumination all matter; adding arbitrary particles does not guarantee activation.

Removal of reactive nitrogen from an air mass through formation and sedimentation of nitric-acid-containing particles. This can delay reformation of chlorine nitrate and prolong active chlorine chemistry. It is a physical/chemical atmospheric removal process, distinct from biological denitrification.

Here $s={\sinh }^{2}r$. The pair moment is $⟨N(N-1)⟩=3s^2+s$. These expressions follow from the single-mode Bogoliubov transformation or by differentiating the even-occupation probability generating function $G(t)=[\cosh r\sqrt{1-t^2{\tanh }^{2}r}{]}^{-1}$.

For a classical ideal gas in a three-dimensional harmonic trap, the peak phase-space density is $\mathcal{D}=N(\hslash \bar{\omega }/k_{B}T{)}^3$. At fixed trap frequencies, atom loss accompanied by $T\propto N^{\alpha }$ increases $\mathcal{D}$ only for $\alpha >1/3$.

The upper endpoint of a two-liquid coexistence region in a temperature-composition diagram. In the ordinary upper-critical case, heating above this temperature removes liquid-liquid demixing.