For a chosen stationary time-translation Killing vector field $k$, this is the region where $k$ is spacelike and observers following its orbits cannot be timelike. In an asymptotically flat geometry, the usual choice is the nonrotating time translation at infinity. The stationary-limit boundary has $g(k,k)=0$, but need not be an event horizon. The Kerr ergoregion is a particular rotating-black-hole example: its outer boundary touches the horizon only on the axis for nonzero subextremal spin.

To count nonnegative tuples with ${\sum }_{i=0}^r{n}_i=N$ and $n_i<a_i$, mark violations $n_i\ge a_i$. An intersection of violations indexed by $J$ is in bijection with unrestricted tuples of sum $N-{\sum }_{i\in J}{a}_i$, by subtracting the bounds from those coordinates. Stars and bars counts this intersection; the inclusion-exclusion principle then gives the displayed formula. A negative remaining sum contributes zero.

For a real scalar triplet with $V=\frac{1}{2}{m}^2{\Phi }^2+\frac{\lambda }{4}({\Phi }^2{)}^2$, $\lambda >0$ and $m^2<0$, the vacuum sphere has radius $v^2=-m^2/\lambda$. At ${\Phi }_0=v{e}_3$, the Hessian is $\mathrm{diag}(0,0,2\lambda v^2)$. Global $O(3)\to O(2)$ breaking therefore gives two physical massless Goldstone bosons and one radial scalar with squared mass $2\lambda v^2$. The angular modes cannot be gauged away in a theory with only global rotations.

A duad is an unordered pair of distinct points. On a six-point set there are $\left(\genfrac{}{}{0ex}{}{6}{2}\right)=15$ duads. They are the edges of the complete graph and are grouped into synthemes.

A bijection from the points of one six-set to the totals of another extends by incidence to bijections from duads to synthemes, synthemes to duads, and totals to points. A duad maps to the intersection of its two point-totals. The same duality exchanges internal duads of a three-versus-three partition with cross synthemes of its corresponding partition. The latter rule follows by splitting the six cross perfect matchings into parity classes of three, yielding two triangles on the six totals.

A family of size-$\kappa$ subsets of a set is almost disjoint at $\kappa$ when distinct members have intersection of size less than $\kappa$. Graphs of a long chain under eventual domination give such a family of size ${\kappa }^{+}$ on $\kappa \times \kappa$. Transport by a bijection gives a family on $\kappa$ itself.

Consider the real family $s^4+2\gamma s^3+(K+{\gamma }^2)s^2+2L\gamma s+P{\gamma }^2$ with fixed $K,L,P$. The Routh-Hurwitz criterion reduces its nontrivial determinant to $4{\gamma }^2[L(K-L)+{\gamma }^2(L-P)]$. Strict decay for every finite $\gamma >0$ is equivalent to the displayed inequalities: the constant and slope must be nonnegative, with at least one positive, and coefficient positivity requires $P>0$. The stricter chain $K>L>P>0$ is sufficient but excludes valid equality cases. Neither equality case guarantees a uniform decay margin as $\gamma$ tends to zero or infinity. At $\gamma =0$ there is no asymptotic attraction.

For uniform angular velocity $\omega$, potential $\varphi$, and additional forces doing no work in the rotating frame,The Coriolis force does no work; the centrifugal potential accounts for the remaining apparent force. This energy is conserved precisely when the potential has no explicit time dependence in the rotating coordinates. An inertial time-independent conservative force need not meet that condition unless its potential is also invariant under the frame's rotation. The balance prevents applying a rotating-frame conservation law to a nonaxisymmetric stationary inertial potential.

The rigid-lid approximation removes external surface displacement from volume conservation while retaining its pressure as a constraint multiplier. It does not mean zero horizontal pressure gradient. In two-layer geostrophic flow, $f_0{\psi }_1=\Pi$ and $f_0{\psi }_2=\Pi -g\prime \chi$, so the upper-layer depth anomaly is $\chi =f_0({\psi }_1-{\psi }_2)/g\prime$. This pressure difference supplies internal potential-vorticity coupling.

Two equal uniform rods of length $2l$ and mass $M$, one pivoted to the floor and the other endpoint sliding on it, have arch coordinates $B=(2l\cos \theta ,2l\sin \theta )$ and $C=(4l\cos \theta ,0)$. The kinetic energy decomposition about the center of mass gives $T=\frac{4}{3}M{l}^2(1+3{\sin }^2\theta ){\dot{\theta }}^2$, while gravitational potential energy is $2Mgl\sin \theta$. Conservation of energy from rest at angle $\alpha$ gives the displayed equation and hinge impact speed $\sqrt{6gl\sin \alpha }$, provided contact is maintained. The endpoint's normal reaction becomes negative during this constrained fall if $\sin \alpha >35/54$; a unilateral floor then permits detachment and the constrained formula no longer describes the entire motion.

In dimension three the Riemann curvature tensor is $R_{ijpq}=g_{ip}{S}_{jq}-g_{iq}{S}_{jp}-g_{jp}{S}_{iq}+g_{jq}{S}_{ip}$, where $S$ is the Schouten tensor. Consequently every three-dimensional Ricci-flat Riemannian manifold is a flat Riemannian manifold. The implication fails in higher dimensions.

A systematic curved pattern in regression residuals plotted against fitted values suggests a misspecified conditional mean. Positive residuals at both ends and negative residuals in the middle motivate trying quadratic regression. The diagnostic does not by itself identify the error probability distribution or prove heteroscedasticity.

For a positive integer $Q$, the Farey sequence ${\mathcal{F}}_Q$ is the increasing list of reduced fractions in $[0,1]$ with denominators at most $Q$, including $0/1$ and $1/1$. For example,Each denominator $q\ge 2$ contributes exactly $\phi (q)$ new interior fractions, because the allowable numerators are precisely those coprime to $q$. Therefore the number of terms is $1+{\sum }_{q=1}^Q\phi (q)=1+\Phi (Q)$, using the summatory totient function.

For two distinct fractions $a/b<c/d$ in the list, their difference is $(bc-ad)/(bd)\ge 1/Q^2$, since the integer $bc-ad$ is positive. This arithmetic separation, rather than arbitrary real spacing, makes Farey fractions useful in rational approximation. The Stern-Brocot tree organizes the same positive reduced fractions by mediant insertion instead of a denominator cutoff.

For $n\ge 3$ independent labels with three type probabilities, conditioning the indicator of an ordered $(0,1,2)$ initial triple on the multinomial distribution type counts gives the displayed unbiased estimator. The Rao-Blackwell theorem reduces its variance strictly when all three probabilities are positive. At a boundary parameter with a missing type, both variances vanish.

An unbiased finite-variance estimator that is a function of a complete sufficient statistic is the unique uniformly minimum-variance unbiased estimator, up to almost sure equality. Rao-Blackwellization supplies variance reduction, and completeness supplies uniqueness. The theorem applies to a conditional statistical experiment as well, when unbiasedness, sufficiency and completeness are all evaluated under that conditional family.

For a colouring $\chi :[S{]}^r\to \{0,\dots ,k-1\}$ of the $r$-element subsets of a set $S$, a subset $H\subseteq S$ is homogeneous if all its $r$-element subsets receive the same colour. The infinite Ramsey theorem for r-sets guarantees an infinite homogeneous subset when $S=\mathbb{N}$, $r$ is positive and the number of colours is finite. This does not guarantee a computable set when the colouring is computable.

A Bogoliubov quasiparticle is an excitation created by the diagonal mode operator obtained through a Bogoliubov transformation. Its vacuum can contain particles in the original modes because creation and annihilation operators mix. Examples include antiferromagnetic spin waves, Bose-gas phonons and paired-fermion excitations.

For one-dimensional quantum scattering, $t$ is the complex amplitude of the outgoing transmitted wave relative to the incoming wave. The transmission probability is $|t{|}^2$ times the ratio of transmitted and incident flux velocities; with equal asymptotic dispersion it is $|t{|}^2$. A reflectionless potential has $r=0$ and a unit-modulus transmission amplitude whose phase shifts continuum quantization. The supersymmetric factorization of the one-soliton potential gives $t(k)=(k+im)/(k-im)$ for its single-bound-state potential.

The thermal rotor path integral traces paths closed modulo $2\pi$. Lift them to the line and sum endpoints differing by $2\pi m$. In inverse-energy time $0<\tau <\beta$, the kinetic exponent is $I\int {\dot{\varphi }}^{2}d\tau /(2{\hslash }^2)$. Integer $m$ counts trajectory windings, not angular-momentum eigenstates.

A chain with local rotor inertia and a phase-locking interaction $-J\cos ({\varphi }_{n+1}-{\varphi }_n)$. Its smooth-phase limit is an elastic phase field with inertia $I$ and stiffness $J$. Compact winding and phase-slip processes remain part of the full model even when absent from the Gaussian smooth-sector approximation.