On a complete nonsingular spatial quotient with no inner boundary and standard asymptotic flatness, a bounded harmonic function tending to zero on all ends is zero. The static vacuum conformal spatial metric is then Ricci flat; three-dimensional curvature from the Ricci tensor makes it flat. Completeness and an ordinary Euclidean asymptotic end exclude nontrivial flat quotients, yielding Minkowski spacetime with the standard global time coordinate.

A spline knot sequence is a nondecreasing list of spline knots, including their multiplicities, that specifies the junctions and smoothness constraints of a spline space. An order-$k$ B-spline uses $k+1$ consecutive knot entries. Its support must have positive width, and repeated knots are interpreted by confluent divided differences or their limiting convention.

For a fixed spline space and admissible interpolation sites, this linear operator sends a continuous function to its unique interpolating spline. Here $Rf$ is the vector of sampled values and $A$ is the B-spline collocation matrix. It reproduces every spline in its range. Its sensitivity to perturbing the data is quantified by the B-spline interpolation operator norm.

A pressureless bound spherical shell first expands, reaches $\dot{R}=0$ at its maximum radius, and recollapses. In the cycloidal spherical-collapse model, turnaround occurs at $\theta =\pi$, with $R_{\max }=2A$ and $t_{\max }=\pi B$. Its mean density is $9{\pi }^2/16$ times the background density in an Einstein-de Sitter universe.

For a homologous, isolated collapsing sphere, write its Newtonian gravitational potential energy as $U=-aG{M}^2/R$. At spherical-collapse turnaround the coherent kinetic energy vanishes. Conserving total energy and applying the virial theorem to the final equilibrium gives $U_{\mathrm{ta}}=U_{\mathrm{vir}}/2$ and hence $R_{\mathrm{vir}}=R_{\mathrm{ta}}/2$. If the gravitational structure coefficient changes, the ratio is $a_{\mathrm{vir}}/(2a_{\mathrm{ta}})$ instead. Mass loss, surface pressure, external tides and energy exchange also change this estimate.

For spherical collapse in an Einstein-de Sitter universe, take $y=R/R_{\mathrm{ta}}$ and $\tau =H_{\mathrm{ta}}t$, with ${\Delta }_{\mathrm{ta}}=(3\pi /4{)}^2$. The expanding branch has $\tau ={\Delta }_{\mathrm{ta}}^{-1/2}{\int }_0^y[u/(1-u){]}^{1/2}du$. Its small-radius expansion is $\tau =(8/9\pi )y^{3/2}(1+3y/10+O(y^2))$, and its turnaround time is $2/3$. These constants are linked: using another leading coefficient without changing the time normalization no longer gives a density contrast approaching zero at early times. The collapsing branch is $4/3-{\tau }_{\mathrm{expand}}(y)$.

An absorption line is a deficit of observed spectral specific intensity relative to the surrounding continuum. In a photosphere with temperature falling outward, increased line opacity makes photons emerge from cooler layers than the continuum. For an absorption-dominated LTE source, the Planck function then supplies a smaller line specific intensity. A foreground layer with source below the incident continuum gives another direct transfer description.

An emission line is a localized excess of specific intensity above the continuum. Heated chromospheric material can give a line source above the photospheric continuum source. Optically thin circumstellar gas can also add line photons through recombination or collisional excitation. Such line formation need not obey a thermal photospheric source prescription.

For the positive Laplace-Beltrami operator on a closed compact Riemannian manifold, the heat trace is $Z_M(t)={\sum }_j{e}^{-t{\lambda }_j}={\int }_M{K}_M(t,x,x)dV(x)$. It records every eigenvalue with multiplicity. In a finite free isometric quotient it equals $|U{|}^{-1}{\sum }_{u\in U}{\int }_N{K}_N(t,x,ux)dV(x)$. The orbital integrals form a class function of the ambient finite isometry group, so Gassmann equivalence gives equal quotient heat traces and hence Sunada theorem.

For a nearly radial Kepler orbit with pericentre $q$, specific angular momentum is $h\simeq \sqrt{2GMq}$. Relative to a circular reference orbit of radius $a_p$ and mean motion $n_p$, the instantaneous azimuthal rate changes from faster to slower at $r_x=\sqrt{h/n_p}\simeq a_p(2q/a_p{)}^{1/4}$. In a rotating reference frame the polar angle has a turning point at this radius. The true anomaly there differs from $\pi$ by approximately $2\sqrt{q/r_x}$.

The SO(4) group consists of real orthogonal four-by-four matrices of determinant one. It preserves orientation and the Euclidean metric. Its Spin(4) double cover identifies it with the quotient of two copies of SU(2) by their simultaneous central sign.

The space of test functions has the locally convex inductive-limit topology of the spaces of smooth functions supported in a fixed compact set $K$. Each ${\mathcal{D}}_K$ uses seminorms $p_m(\phi )={\max }_{|\alpha |\le m}\Vert {\partial }^{\alpha }\phi {\Vert }_{\infty }$. Sequential convergence means a common compact support and uniform convergence of every derivative. A linear functional on this space is a distribution exactly when its restriction to each ${\mathcal{D}}_K$ satisfies a finite order of a distribution estimate; the order and constant may depend on $K$.

At latitude $\varphi$ and solar declination $\delta$, integrate $S_0\max (0,\sin \varphi \sin \delta +\cos \varphi \cos \delta \cos H)$ over hour angle $H$. Sunset occurs at $H_0=\arccos (-\tan \varphi \tan \delta )$ when this lies in $[0,\pi ]$, with clipping for polar day or night. The displayed daily mean is at the top of the atmosphere; local absorbed heat flux also includes transmission and surface albedo.

A strict preference order is single-peaked on an axis when moving away from its most preferred alternative along either one side of the axis makes alternatives less preferred. Comparisons between opposite sides need not follow symmetric physical distance. A profile is single-peaked when all voters' orders share such an axis. This restricted domain admits the median voter rule, unlike the unrestricted-domain conclusion of the Gibbard-Satterthwaite theorem.

A Condorcet winner defeats every other alternative by a strict majority in pairwise voting. It is unique when it exists. For an odd electorate with single-peaked preferences, the median peak is a Condorcet winner: alternatives on either side lose to the majority whose peaks lie at or beyond the median in the other direction. With an even electorate ties may prevent a strict winner.

An exposed tip receives vapour more efficiently than a sheltered surface and can amplify its lead, producing crystal dendrites and side branches. The planar thermal solidification mechanism is an analogy, not a complete vapour-growth model. The hexagonal ice crystal anisotropy selects six preferred directions; an isotropic linear instability alone does not select six arms.

For a variety $X$ over a perfect field, the singular locus is the closed complement of its smooth locus. It consists exactly of points whose local rings are not regular local rings.^[ref]

A normal variety in this setting has no singular points of codimension zero or one. Its generic local rings are fields, and its codimension-one local rings are one-dimensional Noetherian normal domains, hence discrete valuation rings and therefore regular. Any singular locus of a normal variety thus has codimension at least two.

Over an imperfect field, the nonsmooth locus and the nonregular locus can differ. For $k={\mathbb{F}}_p(s)$, the scheme $X=\mathrm{Spec}k[u]/(u^p-s)$ is the spectrum of a field and is regular. After base change adjoining a $p$th root of $s$, its defining equation becomes a $p$th power, so it is not smooth over $k$. The perfect-field qualifier is therefore essential when using normality to exclude nonsmooth points.

Over a perfect field, a point of an algebraic variety is smooth exactly when its local ring is a regular local ring. The smooth locus of a variety is open; its closed complement is the singular locus.

A localized temperature adjustment separates regions with different fluid density, viscosity and Darcy speed. Mass conservation in the moving frame gives the displayed plateau-depth relation. A speed stated as $V_T=\Gamma u_c$ with $u_c$ a Darcy velocity has effective retardation parameter $\varphi \Gamma$. Setting both laboratory-frame mass fluxes equal neglects storage associated with passage of the temperature front.

A buoyant current beneath an overlying low-permeability cap can enter it when current overpressure exceeds the capillary entry threshold. The plateau criterion compares $\Delta \rho gh\cos \theta$ to that threshold. Cold and warm portions can have different depths and mobilities, giving distinct onset injection rates. These onset criteria do not determine the depleted current after leakage begins.