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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.

On a common positive support, a factorization $f={\varphi }_0(x){\varphi }_C(x;\psi ){\varphi }_S(x;\lambda )$ implies ${\partial }_{\psi }{\partial }_{\lambda }\log f=0$. Thus any nonzero mixed derivative prevents this exact separation. For $n$ independent gamma distributed observations with shape $a$ and rate $b$, this mixed derivative is $n/b$, so exact separation of shape and rate into such factors is impossible.

The Heyting implication defines a nucleus on a frame whose sublocale is the open determined by $a$. It is the least nucleus on a frame sending $a$ to $1$.

This nucleus on a frame determines the closed sublocale complementary to the open determined by $a$. It is the least nucleus on a frame taking $0$ to $a$. In the frame of nuclei on a frame, $c(a)$ and the open nucleus $o(a)$ are complements.

For a frame homomorphism $h:B\to A$, nuclei are equivalent when their restrictions to $h(B)$ agree. The least representative of the class of $j$ isThe least representatives form a frame generated by $c(A)$ and $o(h(B))$. To see minimality, each summand is below $j$ and its value at $hb$ is $j(hb)$; any equivalent nucleus therefore bounds every summand. This construction gives the pushout of $B\to N(B)$ along $h$.

A translation-invariant connected generating functional with a constant external source is extensive in spacetime volume ${\mathcal{V}}_d$. Its total logarithm is dimensionless, while its density has mass dimension $d$. Consequently its dimensionless representation at fixed physical volume must retain ${\mu }^d{\mathcal{V}}_d$. For a free massive scalar with source coupling ${\mu }^{-ϵ/2}h\varphi$, completing the square gives $F={\mathcal{V}}_d{\mu }^{-ϵ}{h}^2/(2m^2)$. This already disproves a literal rewriting of the full uniform-source logarithm using only $\lambda ,m^2/{\mu }^2,h/{\mu }^3$ at fixed volume. In infinite volume the total functional is generally infinite; use the density, a source of compact support, or an explicit volume regulator. After writing the density as ${\mu }^{d}f$, its renormalization group equation has the engineering term $+df$. Ultraviolet subtraction alone does not establish infrared finiteness for a massless constant source.

Because $f(z)/z$ is holomorphic and nowhere zero on the unit disc, its square root with value one at zero exists. Defining $h(z)=z\sqrt{f(z^2)/z^2}$ gives an odd normalized univalent function. If $h(z)=h(w)$, squaring gives $z^2=w^2$; the possibility $z=-w$ and oddness force both to be zero. Applying the area theorem for univalent functions to $1/h(1/\zeta )$ yields $|a_2|\le 2$.

The odd square-root transform of a normalized univalent function has reciprocal exterior expansion $1/h(1/\zeta )=\zeta -(a_2/2){\zeta }^{-1}+\cdots$. Its first negative Laurent coefficient has modulus at most one by the area theorem for univalent functions, proving the bound. If $w$ is omitted by $f$, the normalized map $wf/(w-f)$ has second coefficient $a_2+1/w$. Applying the bound twice gives $|1/w|\le 4$, the Koebe quarter theorem.

A finite group is p-nilpotent when it has a normal p-complement. This terminology is distinct from having a normal Sylow p-subgroup. A finite group is nilpotent exactly when it is p-nilpotent for every prime dividing its order.

An antinode is a position of maximum oscillation amplitude in a standing wave. For $u_n=A\sin (n\pi x/L)\cos ({\omega }_{n}t+\phi )$, the antinodes occur atwhere the amplitude is $|A|$. They sit halfway between consecutive nodes of a standing wave. For the first normal mode, the sole antinode is $x=L/2$. It reaches zero displacement at a quarter oscillation period while moving fastest, which illustrates why an antinode is defined by oscillation amplitude rather than instantaneous height.