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Boyer-Lindquist coordinates adapt the Kerr metric to its stationary and axial symmetries. The horizon radii are the roots $r_{\pm }=M\pm \sqrt{M^2-a^2}$ of $\Delta =r^2-2Mr+a^2$.

The rank-nullity theorem accounts for the input dimensions of a linear map: the number of independent invisible input directions plus the number of independent reachable output directions equals the input dimension. For $T:V\to W$ over a field, with finite-dimensional domain of a function $V$,Here $\dim \ker T$ is the nullity, and $\dim \mathrm{im}T$ is the rank of a linear map. The kernel lives in the input vector space, while the image of a linear map lives in the output vector space. For an $m$-by-$n$ matrix of matrix rank $r$, the formula is $n=(n-r)+r$: the number of columns counts input coordinates. ^[ref]

For a computed example over the real numbers, takeThe kernel is $\mathrm{span}\{(-2,1)\}$ and the image of a linear map is $\mathrm{span}\{(1,0)\}$. Each has dimension one, so the two input dimensions split as $2=1+1$. The vector $(3,1)$ can be writtenThe second summand is undetectable: $T(3,1)=T(1,2)$. The picture uses the standard dot product to choose a visible direction perpendicular to the kernel; this is a convenient geometric choice, not an extra hypothesis of the rank-nullity theorem.

This explains a practical tradeoff in a linear system. If $T(x_0)=b$, every other input with output $b$ is $x_0+k$ for $k\in \ker T$. The nullity therefore counts independent freedom in an existing solution; the rank counts independent reachable output directions. In the example, $T(x,y)=(3,0)$ has all solutions $(x,y)=(3-2t,t)$, while $T(x,y)=(3,1)$ has none. The cokernel records this failure of reachability.

For an $m$-by-$n$ matrix, injectivity requires matrix rank $n$, and surjectivity requires matrix rank $m$. When $m\ne n$, these conditions differ. For example, $B:{\mathbb{R}}^3\to {\mathbb{R}}^2$ with $B(x,y,z)=(x+2y,0)$ has rank one and nullity two, because $\ker B=\mathrm{span}\{(-2,1,0),(0,0,1)\}$. The input accounting is $3=2+1$, although only one of two output directions is reached.

Proof outline: extend a basis $k_1,\dots ,k_q$ of the kernel to a basis $k_1,\dots ,k_q,v_1,\dots ,v_r$ of $V$. Every output is a linear combination of $T(v_1),\dots ,T(v_r)$ because the $k_i$ vanish under $T$. If a linear combination of the $T(v_j)$ is zero, the corresponding combination of the $v_j$ lies in the kernel; linear independence of the full basis forces all its coefficients to vanish. Thus the added basis vectors give a basis of the image of a linear map. Counting them proves the formula.

A circular Kepler orbit at radius $r$ has speed $v_K=\sqrt{\mu /r}$ and mean motion $n=\sqrt{\mu /r^3}$.

The specific angular momentum is angular momentum per unit mass, $\mathbf{h}=\mathbf{r}\times \mathbf{v}$. For a planar Kepler orbit its magnitude is $h=r^2\dot{\theta }=\sqrt{\mu a(1-e^2)}$.

The specific orbital energy is the mechanical energy per unit mass,Its sign distinguishes bound elliptic, marginal parabolic, and unbound hyperbolic Kepler orbits.

An orbital element is one of a set of parameters specifying an orbit at a chosen epoch. For a Kepler orbit, the semi-major axis and eccentricity determine its size and shape, while angular elements determine its orientation and the body's position.

For an ellipse of eccentricity $e$, the semi-minor axis is $b=a\sqrt{1-e^2}$.

This is one of the semi-major and semi-minor axes; the two-axis geometry is the broader concept.

The semi-major axis is one half of an ellipse's longest diameter. For a Kepler ellipse with apsidal distances $r_1,r_2$, it is $a=(r_1+r_2)/2$.

This is one of the semi-major and semi-minor axes; the two-axis geometry is the broader concept.

The apoapsis is the point of an orbit farthest from its force centre.

The periapsis is the point of an orbit nearest its force centre.

A positive-energy inverse-square orbit is a hyperbola with two scattering asymptotes.

Orbital eccentricity distinguishes circular, elliptic, parabolic, and hyperbolic conics.

Binet’s equation rewrites a central-force orbit using $u(\theta )=1/r$. If $\ell =m{r}^2\dot{\theta }$ is angular momentum and the potential is $V(r)$, then

For a rapidly decreasing KdV potential and a normalized bound-state eigenfunction, integration of the Wronskian identity over the real line gives ${\dot{\lambda }}_n=0$. Thus every discrete eigenvalue ${\lambda }_n=-{\kappa }_n^2$ is constant.

Jeans instability is the gravitational growth of density perturbations whose self-gravity overcomes pressure support. For physical sound speed $c_s$, the comoving threshold isModes with $k\ll k_J$ grow gravitationally, whereas modes with $k\gg k_J$ undergo pressure-supported acoustic oscillations.

For observations $X_1,\dots ,X_n$, the unbiased sample variance isFor independent identically distributed observations with finite variance ${\sigma }^2$, it converges in probability to ${\sigma }^2$.

For real periodic KdV data, translation by one period acts on a conjugate basis of scattering solutions byLax compatibility gives $\dot{T}=[\Lambda ,T]$ for the matrix of ${\partial }_t+A$ on the eigenspace. Hence $\mathrm{tr}T=2\mathrm{Re}a$ is conserved.

Ifthen every positive power has constant trace:The final equality is the cyclic property of the matrix trace.

A singularity is non-isolated when every punctured neighbourhood contains another singularity. An accumulation point of poles or essential singularities cannot itself be classified as a removable singularity, pole, or isolated essential singularity.

An isolated singularity is removable, a pole, or essential according as its Laurent principal part has zero, finitely many nonzero, or infinitely many nonzero terms.