A subcover of an open cover $\{U_i:i\in I\}$ of a subset $K$ is a selection of its members whose union still contains $K$. The selected open sets retain their original sizes; selecting a subcover does not mean replacing them by smaller open sets.

Use the Minkowski generating functional $Z=e^{iW}$, source $+J\varphi$, mean field $\phi =\delta W/\delta J$, and quantum effective action $\Gamma =W-\int J\phi$. Then $W^{(2)}=i{G}_2^c$, $W^{(3)}=-G_3^c$, and ${\Gamma }^{(2)}=-(W^{(2)}{)}^{-1}$. Differentiating this inverse gives ${\Gamma }^{(3)}=(W^{(2)}{)}^{-1\otimes 3}{W}^{(3)}=-i(G_2^c{)}^{-1\otimes 3}{G}_3^c$, with integration over the repeated spacetime arguments. Thus amputation of external propagators gives $i{\Gamma }^{(3)}$ in this convention. The full connected two-point functions, rather than bare propagators, perform the exact amputation.

Let $H$ be any antiderivative of the initial velocity, $H^{\prime }=g$. One choice of profiles isOpposite additive constants in $F$ and $G$ cancel, so the displacement is independent of the chosen antiderivative. Even for compact support of $g$, these separate profiles can approach nonzero constants; cancellation in their sum is essential to support widening for the wave equation.

For a short derivation, the characteristic coordinates $\xi =x-ct$, $\eta =x+ct$ give $u_{tt}-c^2{u}_{xx}=-4c^2{u}_{\xi \eta }$ by the chain rule. Thus $u=F(\xi )+G(\eta )$, and the Cauchy data imply $F+G=f$ and $-c{F}^{\prime }+c{G}^{\prime }=g$, giving the profiles above.

These characteristic curves are straight when wave speed $c$ is constant. A right-moving component is constant on $x-ct=s$, and a left-moving component on $x+ct=s$. The full superposition generally is not constant on either family, because the other component can vary along it. Their backwards intersections with the initial line select the endpoints used in D'Alembert's formula.

A conditional odds ratio can differ from the marginal odds ratio after averaging over a predictor, even without confounding. If conditional risks are $p_a(x)$, averaging gives ${\bar{p}}_a=\int p_a(x)dF(x)$, and odds of ${\bar{p}}_a$ are generally not averages of the conditional odds. Consequently a common conditional odds multiplier need not equal the marginal odds multiplier. However, if both groups use the same distribution $F$ and $p_1(x)>p_0(x)$ everywhere, then ${\bar{p}}_1>{\bar{p}}_0$ and the marginal odds ratio is still above one. Noncollapsibility alone cannot reverse a uniformly signed effect under an identical background distribution.

This range lies above both the Kolmogorov length scale and the scalar diffusion cutoff, but below the forcing scale. Molecular viscosity and scalar diffusion have negligible direct influence on transfer at those scales. The Obukhov-Corrsin theory predicts a two-thirds scalar structure function there.

For scalar diffusivity much smaller than kinematic viscosity, a scalar has structure below the Kolmogorov length scale while the velocity is smooth. The strain rate $s\sim (ϵ/\nu {)}^{1/2}$ transfers scalar variance, giving $d{S}_C/dr\sim {ϵ}_c/(sr)$. Integration produces a logarithm plus an additive matching constant.

For scalar diffusivity much larger than kinematic viscosity, scalar diffusion dominates at lengths between the Kolmogorov and scalar cutoff scales. The velocity may still have inertial-range variations, but the scalar is smooth to leading order. The leading isotropic scalar structure function is quadratic, with coefficient fixed by the mean-square gradient and the scalar dissipation rate, rather than by a continuing inertial-convective two-thirds law.

A cross-sectional study measures different individuals during one observation period. Differences between ages can reflect confounding or birth-cohort differences as well as changes within individuals; a longitudinal study is needed to measure the latter directly.

A longitudinal study observes the same individuals repeatedly. Within-person changes distinguish temporal development from persistent differences between people, although calendar-period changes, selective dropout and measurement error still require attention. A random-intercept linear mixed model can represent persistent individual differences and correlated repeated measurements.

A case–control study selects subjects according to an outcome: cases have the outcome and controls represent an appropriate comparison population. It compares their prior exposures or genetic variants to study association. Its retrospective sampling must be accounted for when interpreting risks; selection bias and confounding can distort an unadjusted association. In genetics, matched family pseudo-control genotypes can replace unrelated controls for a conditional transmission disequilibrium test.

The Hamiltonian operator represents the total energy and generates unitary time evolution through the Schrödinger equation.

A shock polar is the locus of possible downstream velocity vectors, or an equivalent pressure/deflection relation, as the orientation of an oblique shock varies for fixed upstream conditions. Entropy admissibility selects physical compressive shocks. The strong-shock perfect-gas velocity polar is a circle, while finite-Mach polars have a different shape.

A rotation-anisotropy diagram compares ordered rotation divided by velocity dispersion with a galaxy's flattening. An oblate isotropic rotator provides a reference curve. Falling below it can indicate anisotropic random support, but inclination, measurement apertures, and the definition of rotation change the interpretation. Intrinsic axis ratios and projected ellipticities are different observables.

For the unit-sphere nonlinear sigma model with action $\int d^{D}x|\nabla \sigma {|}^2/(2g)$, parameterize a slowly rotating background by $n_0=(\cos \theta ,\sin \theta ,0,\dots )$ and tangent vectors $e_1=(-\sin \theta ,\cos \theta ,0,\dots )$, $e_a$ for $a\ge 2$. Write $\sigma =n_0\sqrt{1-{\pi }^2}+{\sum }_{a=1}^{n-1}{e}_a{\pi }_a$. For constant slow $\nabla \theta$, the coefficient of $|\nabla \theta {|}^2$ in $|\nabla \sigma {|}^2$ is $1-{\sum }_{a=2}^{n-1}{\pi }_a^2$. The fast Gaussian field theory gives $⟨{\pi }_a^2⟩=g{I}_D$, where $I_D={\int }_{\Lambda /b<|p|<\Lambda }{d}^{D}p/[(2\pi {)}^D{p}^2]$. The omitted linear term integrates to zero, and terms cubic in the fast variables have zero one-loop average. Thus precisely $n-2$ tangent components reduce the phase stiffness $1/g$. Restoring the cutoff in $D=2+ϵ$ gives $dg/d\log b=-ϵg+(n-2)S_D{\Lambda }^{ϵ}{g}^2/(2\pi {)}^D+O(g^3)$. This weak-coupling expansion requires $n\ge 2$; the discrete target $S^0$ has no tangent expansion.

For a scalar conservation law whose flux is a convex function, the Godunov numerical flux is the physical flux at the origin of the entropy solution of the Riemann problem between states $a,b$. It is ${\min }_{u\in [a,b]}f(u)$ for $a\le b$ and ${\max }_{u\in [b,a]}f(u)$ for $a>b$. For a rarefaction wave, the characteristic speed is monotone across its states, so the origin sees either an endpoint or a zero-speed minimizing state. For a shock wave, its speed is $(f(a)-f(b))/(a-b)$; the origin sees the left state if this speed is positive and the right state if negative, selecting the larger endpoint flux. Unlike the Godunov flux, the Engquist-Osher flux at reversed transsonic states may equal $f(a)+f(b)-f(s_{\ast })$.

A Lebesgue null set has zero Lebesgue measure. Since Lebesgue measure is complete, every subset of one is also measurable and null. The Cantor set is an uncountable example with positive Hausdorff dimension; the rational numbers form a countable, dense example with Hausdorff dimension zero. Being null does not imply being empty or having finitely many points.

In four spacetime dimensions, for an affinely parametrized twist-free null geodesic congruence, let $A$ be local transported screen area and ${\sigma }^2={\hat{\sigma }}_{AB}{\hat{\sigma }}^{AB}/2$. The Null Raychaudhuri equation gives ${\theta }^{\prime }=-{\theta }^2/2-2{\sigma }^2-R_{ab}{k}^a{k}^b$ and $A^{\prime }/A=\theta$. Combining these gives the displayed equation. Positive shear focuses the beam and has a negative sign. For nonuniform expansion this is a local area identity, not the same differential equation for the entire integrated horizon area.

In the exterior of Schwarzschild spacetime, $d{r}_{\ast }/dr=(1-2M/r{)}^{-1}$. Outgoing radial null geodesics therefore have constant $u=t-r_{\ast }$. At fixed receiver radius $R$, their arrival Schwarzschild time is $u+r_{\ast }(R)$, so differences in $u$ give arrival-time differences. A static receiver's proper time includes the factor $\sqrt{1-2M/R}$.

// Target: optics.bigb

Contract the Einstein field equations with a null vector. Both the trace term and the cosmological constant term multiply $g_{ab}{\ell }^a{\ell }^b=0$ and disappear. The null energy condition therefore still implies the null convergence condition, independently of the sign of the cosmological constant. A black-hole area theorem still needs the appropriate global horizon assumptions as well as this local focusing condition.