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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 $e$ measures the shape of a Kepler orbit. A circle has $e=0$, an ellipse $0<e<1$, a parabola $e=1$, and a hyperbola $e>1$ for a nonradial attractive orbit.

For a bound ellipse with semi-major axis $a$, pericentre $r_p=a(1-e)$ and apocentre $r_a=a(1+e)$ giveThus unequal turning distances measure eccentricity directly. In a perturbed orbit, osculating orbital elements can change with time; eccentricity need not remain conserved.^[ref]

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.

Let $L(t),A(t)$ be differentiable finite square matrices satisfying the Lax equation $\dot{L}=[L,A]$. For each positive integer $n$, the product rule and cyclicity of the matrix trace giveThus each power trace is an integral of motion. The commutator order agrees with the stated evolution equation; reversing both orders reverses the flow but preserves the conclusion. For infinite-dimensional operators, the traces must exist and cyclicity must be justified; the finite-matrix argument alone does not establish those analytic hypotheses.

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.

Suppose $f$ is holomorphic on $0<|z-a|<r$. Its Laurent series isThe principal part consists of terms with $n<0$. If all these coefficients vanish, the remaining power series extends across $a$, so the singularity is a removable singularity. If the lowest nonzero power is $-m$ for a positive integer $m$, thenwith $h$ holomorphic. This is a pole of order $m$. Infinitely many nonzero negative coefficients define an essential singularity.

These cases are mutually exclusive and exhaustive: a nonempty finite set of negative exponents has a least member, while an infinite such set is unbounded below. The Laurent series is unique, so it cannot support two different types. For $\sin z/z$, $1/z^3$ and $e^{1/z}$ at zero, the types are respectively removable, a pole of order three and essential. This classification requires the singularity to be isolated; the presence of arbitrarily nearby additional poles invalidates the punctured-disk hypothesis.

The Zariski topology on affine space is Noetherian because polynomial rings are Noetherian. Every closed subset is a finite union of irreducible closed subsets: a minimal counterexample would split into two smaller closed subsets, each already having such a decomposition.

An irreducible component is a maximal irreducible closed subset. Every algebraic variety is a finite union of its irreducible components.

For $D_{2n}=⟨r,s:r^n=s^2=1,srs=r^{-1}⟩$, every irreducible complex representation has degree at most two. If $v$ is an eigenvector of $r$ with eigenvalue $\lambda$, then $sv$ has eigenvalue ${\lambda }^{-1}$, so irreducibility makes the representation equal to $\mathrm{span}\{v,sv\}$.

If conjugation sends an operator to its inverse, the conjugating element sends its $\lambda$-eigenspace to the ${\lambda }^{-1}$-eigenspace.

For a monic irreducible separable polynomial $g\in K[T]$ in characteristic $p$, $g(T^p)$ is reducible exactly when every coefficient of $g$ is a $p$th power in $K$. If $g=\sum a_i^p{T}^i$, thenConversely, if $g(T^p)$ is reducible, all its irreducible factors are separable. Its zero derivative forces every factor multiplicity to be divisible by $p$, so it is a $p$th power in $K[T]$ and all its coefficients are $p$th powers.

The principal inverse sine is the inverse of the sine restricted to $[-\pi /2,\pi /2]$:The range restriction is essential because the sine is not injective on all of the real numbers. In the interior $-1<x<1$, implicit differentiation givesThe positive square root appears because the cosine is positive in $(-\pi /2,\pi /2)$. For example, $\arcsin (\sin (3\pi /4))=\pi /4$, not $3\pi /4$. All real solutions of $\sin \theta =x$ are $\theta =\arcsin x+2k\pi$ or $\theta =\pi -\arcsin x+2k\pi$, with $k$ an integer.

A boundary point with $\dot{V}=0$ can still enter the sublevel set; LaSalle analysis decides whether it belongs to an invariant zero-derivative trajectory.

The Lindeberg replacement method, inspired by the Lindeberg condition, compares functions of two independent random vectors by replacing their coordinates one at a time. Matching moments cancel the corresponding terms in a Taylor expansion.

For prescribed values $F(x_i)$ at ordered points $x_0<\cdots <x_n$, the connect-the-dots interval map is the unique piecewise linear function obtained by linear interpolation between consecutive data points.