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For a twice differentiable position away from the origin, write $\mathbf{r}=r\mathbf{u}$ with radial unit vector $\mathbf{u}$. Its unit-length constraint gives $\mathbf{u}\cdot \dot{\mathbf{u}}=0$ and $\mathbf{u}\cdot \ddot{\mathbf{u}}=-|\dot{\mathbf{u}}{|}^2$, separating radial and directional contributions to speed and kinetic energy. The identity is kinematic and does not require a particular force law.

A degenerate Killing horizon has zero surface gravity. In the radial quotient of an extremal black hole, a double horizon root produces a pole in the tortoise coordinate rather than the logarithm of a simple root. Such a horizon has no bifurcation surface of the usual nondegenerate type.

For a Rademacher sum in a normed vector space, any two fixed positive moments of its norm are comparable with constants depending only on those moments. The sharp Rademacher second-moment inequality gives the sharp first-to-second moment comparison $\Vert S{\Vert }_{L^2}\le \sqrt{2}\Vert S{\Vert }_{L^1}$ for arbitrary coefficients.

The irreducible mass of a four-dimensional uncharged Kerr black hole is $M_{\mathrm{irr}}=\sqrt{A/(16\pi )}$. Its mass satisfies $M^2=M_{\mathrm{irr}}^2+J^2/(4M_{\mathrm{irr}}^2)$. Hawking's area theorem prevents a classical process from decreasing $M_{\mathrm{irr}}$ under the theorem's energy and global hypotheses.

Ingoing Kerr coordinates cancel the explicit inverse-$\Delta$ singularities of the Boyer-Lindquist coordinates metric. They give a smooth, nondegenerate Kerr metric at the future outer horizon when $M>|a|$. Its stationary and axial Killing vector fields are ${\partial }_v$ and ${\partial }_{\chi }$. The advanced chart need not cover every horizon branch of the maximal extension.

The outer Killing horizon of a Kerr black hole is generated by $k+{\Omega }_{H}m$, with the stationary Killing vector field $k$ normalized at infinity and the axial generator $m$ having $2\pi$-periodic orbits. In ingoing Kerr coordinates its generators satisfy $d\chi /dv={\Omega }_H$. This is the horizon's angular velocity relative to the nonrotating asymptotic stationary frame.

For the massless Klein-Gordon equation on a Kerr black hole, the covariant wave operator separates into radial and angular ordinary differential equations. With $K=(r^2+a^2)\omega -am$ and angular separation constant $\Lambda$, the radial equation is $(\Delta R^{\prime }{)}^{\prime }+[K^2/\Delta -a^2{\omega }^2+2am\omega -\Lambda ]R=0$. The angular equation is $(\sin \theta {\Theta }^{\prime }{)}^{\prime }/\sin \theta +[a^2{\omega }^2{\cos }^2\theta -m^2/{\sin }^2\theta +\Lambda ]\Theta =0$. No extra radial factor is included in the displayed ansatz.

An orbit about a star. Dust on such an orbit can experience radiation pressure and Poynting–Robertson drag, in addition to gravitational perturbations.

An orbit about a planet. Stellar tides, radiation pressure and other perturbations can limit its stability; a sufficiently small semi-major axis compared with the Hill sphere permits a local approximately planet-centred Kepler orbit.

For an attractive inverse-square force, $r=p/(1+e\cos f)$, where $f$ is true anomaly, $e$ is orbital eccentricity, and $p=h^2/\mu$ is the semilatus rectum. The specific angular momentum relation $\dot{f}=h/r^2$ converts this geometric equation into an angular evolution law.

In the centre-of-mass frame of two Newtonian point masses, total orbital energy is reduced mass times specific orbital energy: $E=\mu [|\dot{\mathbf{r}}{|}^2/2-GM/r]$. The factor $\mu$ multiplies both kinetic and potential contributions, while $GM$ is the relative gravitational parameter.

For unit mass in $\Phi =-GM/r$, conservation of angular momentum splits the velocity norm into ${\dot{r}}^2+L^2/r^2$. Conservation of energy then gives $\frac{1}{2}{\dot{r}}^2+L^2/(2r^2)=E+GM/r$. It separates radial dynamics from the reconstruction of the orbital angle.

For an instantaneous in-plane change of velocity of magnitude $\gamma v_K$ on a circular orbit, with angle $\theta$ measured from the forward tangential direction, the new specific orbital energy gives $a^{\prime }/a=(1-2\gamma \cos \theta -{\gamma }^2{)}^{-1}$. The specific angular momentum gives $e^{\prime 2}=1-(1-2\gamma \cos \theta -{\gamma }^2)(1+\gamma \cos \theta {)}^2$. The signed semi-major axis becomes negative for an unbound hyperbolic Kepler orbit.

For positive chemical concentration $c$, taking $\chi (c)=\alpha /c$ makes the chemotaxis drift velocity $\chi (c)c_x=\alpha {\partial }_x\log c$. Cells then respond to a relative chemical gradient. This singular response can remain finite where the concentration tends to zero if its logarithmic derivative has a finite limit.

If a censored individual is allocated event weights $q_j$ over later event times, its event-count contribution at time $t_j$ is $q_j$ and its risk set contribution is ${\sum }_{k\ge j}{q}_k$. These fractional counts can be used in the Kaplan–Meier estimator.

The Kaplan–Meier estimator is self-consistent under fractional allocation of censored event times according to its own conditional fitted event distribution. Adding the corresponding fractional event counts and risk set weights leaves the estimator unchanged.

The neutral flavour states $d\bar{s}$ and $s\bar{d}$ can mix through second-order weak interactions. Box diagrams with W bosons and internal up-type quarks induce a strangeness-changing-by-two four-quark operator. The GIM mechanism removes its flavour-independent part.

The transport cost function assigns a cost to a unit of mass moved from $x$ to $y$. A transport plan has total cost $\int cd\pi$. Costs separating as $a(x)+b(y)$ have the same well-defined total for every coupling of fixed marginals.

For $h=\alpha d{x}^2+\beta d{y}^2+\chi (dz-ydx{)}^2$, the Kaluza-Klein decomposition along a Killing field ${\partial }_x$ has $V=\alpha +\chi y^2$, $A=-\chi ydz/V$ and $\gamma =\beta d{y}^2+\alpha \chi d{z}^2/V$. For a Riemannian metric, positive constants make the fibre norm and quotient metric positive everywhere.

For $N\ge 2$, $U_{ij,\theta }F=F\circ R_{ij,\theta }$ uses the genuine two-coordinate rotation $(v_i,v_j)\mapsto (v_i\cos \theta +v_j\sin \theta ,-v_i\sin \theta +v_j\cos \theta )$. Average it uniformly over angles and pairs to obtain $Q$. Rotation invariance of Lebesgue measure makes $Q$ a bounded self-adjoint operator of norm one on $L^2({\mathbb{R}}^N)$. Each pair average is an orthogonal projection, but their full average is not in general a projection.