A steady full-sphere radial inflow has mass flux $\dot{M}=4\pi r^2\rho |v_r|$ by conservation of mass. With Newtonian free-fall speed from rest at infinity, $|v_r|=\sqrt{2GM/r}$, this becomes the displayed expression. The assumed velocity then requires $\rho \propto r^{-3/2}$ along the steady flow. A local density and speed do not determine the total mass accretion rate of a general nonspherical inflow, and this estimate does not impose the sonic boundary condition of Bondi accretion.

Policy iteration alternates evaluation of a fixed decision policy with improvement of its actions using the resulting value function. A stable policy satisfies the Bellman equation when each chosen action attains the required optimum.

The potential scale reduction factor compares variation within simulated chains to a variance estimate also incorporating differences between chains. In its classical form, $\hat{R}=\sqrt{\hat{V}/W}$, where $W$ is the average within-chain sample variance. Values appreciably above one indicate disagreement; values close to one do not prove convergence. Splitting chains helps expose changes over time, and modern rank-based variants improve robustness.^[ref]

For a flux threshold $f_{\min }$ and isotropic point-source luminosity $L$, the detection radius is $r_{\max }=\sqrt{L/(4\pi f_{\min })}$. In a homogeneous Euclidean population the accessible volume is proportional to $L^{3/2}$, so the selected luminosity distribution is proportional to $\phi (L)L^{3/2}$. This assumes flux selection and inverse-square propagation, not an arbitrary surface-brightness-selected diffuse field.

A noise-free flux threshold truncates the distance distribution of objects with exactly identical relevant-band luminosity. It cannot preferentially select intrinsically brighter members when no luminosity scatter exists. A luminosity distribution introduces the accessible-volume weighting $L^{3/2}$ in homogeneous Euclidean space. Measurement scatter across the flux threshold and nonuniform completeness introduce additional selection effects even for identical intrinsic luminosities.

In a uniform ideal-MHD plasma at rest, take the wavevector parallel to a nonzero background field. Projecting the linear velocity equation along and perpendicular to the field gives two transverse Alfvén polarizations at speed $v_A$ and one longitudinal acoustic polarization at speed $c_s$. Compression does not bend field lines for the longitudinal branch. The general oblique fast/slow magnetosonic modes reduce to the corresponding faster/slower parallel branches, with degeneracies when their speeds agree.

Reversal between opposite magnetic orientations through quantum mixing rather than activation over the full classical anisotropy barrier.

A finite molecular spin system exhibiting slow reversal of its magnetic moment on an experimental timescale. Blocking is timescale-dependent and does not require bulk long-range magnetic order.

A ferromagnetic particle with no internal domain wall. Magnetization reversal can proceed by rotation or nonuniform modes; thermal activation and anisotropy determine blocking and coercivity. Small size alone does not guarantee a permanent hard magnet.

For strain $\gamma ={\gamma }_0\cos \omega t$ and stress $\sigma ={\gamma }_0(G^{\prime }\cos \omega t-G^{\prime \prime }\sin \omega t)$, integrating supplied power $\sigma \dot{\gamma }$ over one period gives the displayed dissipated energy per unit volume. The storage term integrates to zero; passivity requires nonnegative loss modulus for positive frequency in this convention.

The radius of a charged particle orbit perpendicular to a uniform magnetic field. In a parabolic band $r_c=m^{\ast }{v}_{\perp }/(|q|B)$, or $r_c=\hslash k_F/(e|B|)$ for a Fermi-level electron. Charge sign sets the curvature direction, while the radius uses magnitudes.

For a square spanning $0\le x,y\le a$, with current counterclockwise as seen from positive $z$, an applied field $B_z=\alpha xy$ vanishes on two edges. Integrating $Id\mathbf{\ell }\times \mathbf{B}$ on the other two gives the displayed force. Its direction reverses with the current. The formula concerns the loop, not unspecified supply leads, and cannot be replaced by a small-loop approximation without checking the field variation.

For a two-dimensional Thomas–Fermi condensate at fixed coupling, $N=\pi n_0{R}^2/2\propto {\mu }^2$, so $R\propto N^{1/4}$. Slow exponential atom loss shrinks the cloud while a fixed stirring radius becomes a larger fraction of its size. At $\Omega ={\omega }_r/\sqrt{2}$ and initial $D/R=1/2$, the local-sound-speed threshold is reached at $t_c=2\tau \ln 2$, before the shrinking cloud edge reaches the beam.

The complementary log-log link maps $p\in (0,1)$ to $\ln [-\ln (1-p)]$, with inverse $p=1-\exp (-e^{\eta })$. For constant infection hazard $\lambda$ and exposure $e$, $p=1-e^{-\lambda e}$ implies $\mathrm{cloglog}(p)=\ln \lambda +\ln e$. Thus a binomial regression with a log-exposure offset estimates the log hazard as its intercept.

The linearization of a Riccati equation turns this equation into $z^{\prime \prime }-(a+b)z^{\prime }+abz=0$. For distinct real roots, every nonzero linearizing solution is $z=A{e}^{ax}+B{e}^{bx}$ and $y=(aA{e}^{ax}+bB{e}^{bx})/(A{e}^{ax}+B{e}^{bx})$ on intervals with nonzero denominator. The constants $y=a,b$ arise when one coefficient vanishes. With $A\ne 0$, the relative coefficient $B/A$ can have either sign; replacing it by a fixed-sign real exponential would omit a real family of solutions.

For $a=b$, the solutions are $z=(A+Bx)e^{ax}$ and $y=a+B/(A+Bx)$. The equilibrium $y=a$ occurs at $B=0$. Zeros of a nontrivial $z$ give poles of a Riccati solution from zeros of its linearizing solution, so the corresponding $y$ is defined separately between them. Multiplying both coefficients by the same nonzero scalar changes $z$ but not $y$.

In time-translation-invariant linear response, each infinitesimal past strain step contributes its amplitude times the elapsed-time relaxation modulus. Adding those causal contributions gives the displayed stress-history integral for shear. An equivalent formulation convolves the creep compliance with stress increments. This constitutive law assumes sufficiently small strains about a specified equilibrium or relaxed state; nonlinear deformation, aging and evolving material structure can invalidate that simple superposition.

A Maxwell fluid branch connected in parallel with a Newtonian dashpot. Branches share strain and their stresses add. With $\tau ={\eta }_M/G$, the displayed modulus uses the $e^{i\omega t}$ convention. Its low-frequency viscosity is ${\eta }_M+{\eta }_s$ and static shear modulus zero, so the material ultimately flows.

For a series Maxwell model with $\tau =\eta /k$, $E^{\prime \prime }=k\omega \tau /(1+{\omega }^2{\tau }^2)$. At fixed strain amplitude the energy dissipated per cycle is $\pi E^{\prime \prime }{\epsilon }_0^2$ and peaks at $\omega \tau =1$. At fixed stress amplitude it is instead $\pi {\sigma }_0^2/(\eta \omega )$, which has no finite-frequency maximum. Specifying the imposed variable and whether loss means energy per cycle or power is essential.

For the linear-in-potential charged-scalar Lagrangian, $\pi ={\dot{\varphi }}^{\ast }-ie{A}_0{\varphi }^{\ast }$ and ${\pi }^{\ast }=\dot{\varphi }+ie{A}_0\varphi$. With $A_{\mu }=(A_0,-\mathbf{A})$, the Hamiltonian density isThe first product includes $+e^2{A}_0^2|\varphi {|}^2$ after the Legendre transform; that term can arise in canonical variables even though the original Lagrangian has no interaction quadratic in $A$.

For $\delta \varphi =-iϵ\varphi$, the conserved current is $J^{\mu }=i({\varphi }^{\ast }{\partial }^{\mu }\varphi -\varphi {\partial }^{\mu }{\varphi }^{\ast })-2e{A}^{\mu }|\varphi {|}^2$. The conjugate field equations implywhich directly cancels the divergence of the explicit background term. Conservation requires no Lorenz gauge condition; constancy of total charge additionally requires vanishing boundary flux.