A uniform overdensity evolves as a separate spherical pressureless region until shell crossing. In an Einstein-de Sitter background, formal collapse has linearly extrapolated threshold ${\delta }_c=(3/5)(3\pi /2{)}^{2/3}\simeq 1.686$. This is distinct from the actual nonlinear density. Virialization and nonspherical tides require further physics.

A model assigning nonoverlapping spheres to particles; contacts express geometric constraints. A crystallographic sublattice need not be physically close-packed even when its Bravais lattice is face-centred cubic.

For forward motion with nonnegative speed $v(t)$ over duration $T>0$ and distance $L={\int }_0^{T}v(t)dt>0$, define the time-averaged speed $V_t=L/T$ and distance-averaged speed $V_x=L^{-1}\int vdx$. The change of variables $dx=vdt$ gives $V_x={\mathbb{E}}_t[v^2]/{\mathbb{E}}_t[v]$, where ${\mathbb{E}}_t$ averages uniformly over time. ThereforeEquality holds exactly for constant speed almost everywhere. This also follows directly from the Cauchy-Schwarz inequality applied to $v$ and one on the time interval. The reason for the difference is that distance averaging puts more weight on periods when the vehicle covers more distance per unit time. These formulas assume a finite second time moment; if that moment is infinite, the distance average is infinite and the inequality holds in the extended sense.

The finite width of a spectral line can arise from finite optical coherence or a distribution of resonance frequencies. lifetime broadening and impact-regime pressure broadening often give Lorentzian line shapes; thermal Doppler broadening gives a Gaussian component. The instrument response must also be included when interpreting a measured spectrum.

A charged fermionic mass eigenstate formed by mixing charged electroweak gauginos and higgsinos in a supersymmetric model. Its production and decay can yield leptons and missing transverse momentum when a stable neutral lightest state is present.

For a three-dimensional parabolic band and fixed positive $\mu -E_c$, the Sommerfeld expansion gives $n(T)=n(0)[1+{\pi }^2(k_{B}T/(\mu -E_c){)}^2/8+O(T^4)]$. This increase differs from the usual fixed-density calculation, in which the chemical potential decreases with temperature. The exact density is proportional to $T^{3/2}{F}_{1/2}[(\mu -E_c)/(k_{B}T)]$.

Reorganization of a solvent to accommodate a solute. Its free-energy cost competes with favourable solute–solvent interactions and depends on solute size and shape.

A ratio of concentrations or activities between specified phases. State whether all protonation states are combined: a pH-dependent total distribution ratio differs from a partition coefficient for one molecular species.

The shape of a solid-state precipitate. Thin plates can result when broad faces have much lower interfacial energy than edge regions.

Evolution toward larger characteristic precipitate sizes, often by diffusional growth of larger particles at the expense of smaller ones. The associated changes of spacing and shape can alter strengthening.

A meridian $r=r(\theta )$ has cylindrical radius $\rho =r\sin \theta$ and height $z=r\cos \theta$. On a branch with monotone height and filled horizontal disk sections, the volume between nearby horizontal planes is $dV=\pi {\rho }^2|dz|$, giving the formula. The conditions matter: a meridian that folds in height or bounds annular sections requires separate branches and inner-radius subtraction. A spherical angular wedge instead has volume $(2\pi /3)r(\theta {)}^3\sin \theta d\theta$ when the solid fills each radial segment. Both describe volumes, but they partition the solid differently.

A trans-SNARE complex links SNARE proteins anchored in different lipid bilayers. Its formation can bring membranes together before complete membrane fusion.

A class of Caenorhabditis elegans secondary small RNAs with a characteristic 5′ guanine and approximately 22-nucleotide length, generated by RNA-dependent RNA polymerases.

For a light nonmagnetic rod of length $l$ wound uniformly with $N$ thin turns of mean radius $a$, the magnetic dipole moment is $m_B=NI\pi a^2$. Wire of cross-sectional area $A_w$ and density $\rho$ has mass approximately $M=2\pi aN\rho A_w$. The transverse central moment of inertia is $J=M(l^2/12+a^2/2)$ in the thin-winding approximation. Consequently the oscillation period in a transverse magnetic field does not depend on the number of turns: wire inertia and magnetic moment both scale with $N$.

Measurements at fixed frequency and varying oscillation amplitude. Amplitude-independent moduli identify a possible linear regime; dependence on amplitude signals nonlinear response or an experimental artifact.

Measurements at different frequencies while retaining an amplitude in the linear regime. They compare observation time with material relaxation times.

For $\rho =k/(2\pi G{r}^2)$ truncated at $r_0$, the enclosed mass is $M(r)=2kr/G$. The shell-assembly integral gives $W=-\int GM(r)dM/r=-G{M}_0^2/r_0$. The origin has divergent density but finite total mass and binding energy.

With potential zero at infinity, a truncated profile $\rho =k/(2\pi G{r}^2)$ has $\Phi (r)=-2k[1+\ln (r_0/r)]$ inside the truncation. Exterior spherical shells have zero force inside but contribute a nonzero constant potential there; omitting them changes escape work.

For $p=a_T^2\rho$, spherical hydrostatic equilibrium admits $\rho =a_T^2/(2\pi G{r}^2)$ with $M(r)=2a_T^{2}r/G$. Matching an external pressure $p_0$ fixes $r_s=a_T^2/\sqrt{2\pi G{p}_0}$. The central density is singular although the enclosed mass tends to zero; this is one branch, not every finite-centred isothermal cloud.

Truncating the singular isothermal gas profile at radius $r_0$ gives $M=2c_s^2{r}_0/G$ and boundary pressure $p_0=c_s^4/(2\pi G{r}_0^2)$. The internal energy of a monatomic gas is $U=3M{c}_s^2/2$, while shell integration gives gravitational self-energy $\Omega =-G{M}^2/r_0=-2M{c}_s^2$. Thus $2U+\Omega =3p_0\mathcal{V}$, with $\mathcal{V}=4\pi r_0^3/3$. The confining pressure supplies the virial surface term. This singular-central-density model differs from a regular pressure-confined isothermal sphere.