A separation axiom is a condition on a topological space expressing how points or closed subsets can be distinguished by open sets. The Kolmogorov space condition distinguishes distinct points by some open set; the T1 space condition makes every point closed; the stronger Hausdorff space condition gives disjoint neighbourhoods to distinct points.

A G-delta set is a countable intersection of open subsets of a topological space. In a complete metric space these are precisely the topologically complete subspaces, by the G-delta criterion for topological completeness.

Undo a short evolution of two orthogonal locally indistinguishable states. Orthogonality is preserved exactly. A Lieb-Robinson bound localizes every backward-evolved observable within an expanded region and bounds its norm error. After normalizing the approximate observable, the initial expectation difference is at most $\epsilon (1+\delta )+2\delta$. Polynomial volume growth in the system diameter controls the localization prefactor and yields preservation for sufficiently small linear-time coefficient.

Invariance of domain says that a continuous map that is an injective function between topological manifolds without boundary of the same dimension is an open map. It is useful in the Penrose singularity theorem: projecting a compact achronal boundary along timelike curves onto a Cauchy hypersurface gives a nonempty subset that is both open and closed, forcing that connected Cauchy hypersurface to be compact.

A connected component, or homotopy class, of admissible field configurations with fixed asymptotic conditions. Distinct topological sectors cannot be joined by a continuous deformation within that configuration space. A nonzero topological charge labels a sector, but a stable finite-size energy minimum also depends on the energy functional.

A local current whose conservation follows from field geometry rather than requiring the field equation. For Skyrmions, pulling back the closed volume form on the target three-sphere gives the current associated with topological baryon number in the Skyrme model. Its spatial integral is a topological charge, unchanged by smooth evolution with the prescribed boundary behaviour.

A nonzero topological degree obstructs smooth unwinding with fixed boundary data, but does not by itself give a stable finite size. In three dimensions a two-derivative energy scales as $R$ and can decrease as a configuration shrinks at fixed degree. The four-derivative Skyrme term scales as $R^{-1}$ and can balance this tendency. Derrick theorem addresses the energetic issue; a singular limiting configuration can evade smooth homotopy conservation.

A normalized vacuum field on a sphere surrounding a core can define $S^{d-1}\to S^{d-1}$ with integer topological degree. Nonzero degree obstructs its extension through the enclosed ball while staying in the vacuum manifold. This accounts for the winding number of a vortex and the spherical degree of a magnetic monopole, with the appropriate vacuum targets.

With common factors of $\pi$ removed, steady kinematic fluxes satisfy $Q_z=2\alpha \sqrt{M}$, $M_z=FQ/M$ and $F_z=-N^{2}Q$. The Batchelor entrainment hypothesis supplies volume, the integrated buoyancy force supplies momentum, and the environmental mass density gradient reduces buoyancy flux. This uses the Boussinesq approximation.

If two equal pure plumes merge while keeping their common velocity and reduced gravity, conservation of cross-sectional area gives radius $\sqrt{2}$ times the individual radius. All three integral fluxes double. The plume balance parameter increases by $2^3/2^{5/2}=\sqrt{2}$, so the immediate combined plume is a lazy plume. Similarity attraction may restore pure balance farther downstream, but instantaneous pure balance does not follow from adding two pure plumes.

If a Fermi interaction contains vector and axial operators with the same time-reversal symmetry component signs, antiunitarity conjugates their coupling constants without exchanging the two operators. After removing a common field phase, invariance requires $g_A/g_V$ real when $g_V\ne 0$. Equivalently $\mathrm{Im}(g_A{g}_V^{\ast })=0$. Fixed intrinsic phases can additionally constrain the common phase; the relative condition is independent of that choice. Adding a Hermitian-conjugate interaction guarantees a Hermitian operator Hamiltonian, but does not guarantee time-reversal symmetry.

The twin paradox compares proper time along the different world lines of a travelling observer and a stay-at-home observer who reunite. Reciprocal inertial time dilation produces an apparent contradiction only if the traveller's change of inertial frame is ignored.

The decisive experiment is local: put two clocks together at departure, separate them, then compare their readings together at reunion. Everyone agrees on those two spacetime events and on the final difference. The twin paradox asks why two clocks can accumulate different proper times between the same spacetime events, despite reciprocal time dilation during uniform motion. Its answer is the geometry of their complete world lines, together with the relativity of simultaneity when comparing distant clocks.

With the Airy displacement integrals, the lower pressure-to-velocity ratio is $-iq\mathrm{Ai}\prime (-qC)/I_0(C)$. The decaying upper mode gives that ratio as $\alpha /\delta$. Hence for $\alpha =k{R}^{-1/4}$ the dispersion relation is $k^{4/3}{I}_0(C)+iq\mathrm{Ai}\prime (-qC)=0$. Writing it without division remains meaningful if an individual moment vanishes. The wavespeed is $c=\delta C$ and a positive wavenumber grows temporally when $\mathrm{Im}C>0$.

A thin circular disc presents projected area $\pi R^2|\cos i|$ to a sightline. Isotropic normals have a uniform distribution of $\cos i$ on $[-1,1]$, giving mean absolute cosine one half. The resulting interception cross-section is therefore half the face-on area; no extra orientation weighting is needed after this average is used in an incidence calculation.

For $b=A(y)+B(z)$, constant nonzero $f_0$, and zero basic velocity at $z=0$, thermal-wind balance givesWith full buoyancy represented by $f_0{\bar{\psi }}_z$, the constant-coefficient three-dimensional quasi-geostrophic potential vorticity is $\bar{q}=f_0+z{A}^{\prime \prime }/f_0+f_0{B}^{\prime }/N^2$. The reference-buoyancy convention in quasi-geostrophic potential vorticity changes only the uniform reference contribution.

Momentum-exchange collisions maintain a common temperature and an equilibrium-shaped momentum probability distribution. Particle abundances and chemical potentials need not have their chemical equilibrium values: elastic scattering alone conserves particle number. Thus kinetic equilibrium does not by itself imply either zero chemical potential or conserved comoving entropy.

Boltzmann suppression is the exponential reduction of an equilibrium population when creating it costs energy much greater than the temperature. For a nonrelativistic species with negligible chemical potential, the leading factor is $e^{-m/T}$, multiplied by its momentum phase-space volume. It concerns the equilibrium abundance; freeze-out can preserve a larger population than the instantaneous equilibrium value at later times.

Here natural units set the Boltzmann constant and the speed of light to one, so the mass $m$ denotes its rest-energy and $T$ has energy units. In ordinary units the exponential function has argument $-m{c}^2/(k_{B}T)$.

A massive decoupled sector enters a cannibal phase when its particles are nonrelativistic but rapid number-changing reactions maintain chemical equilibrium and zero chemical potential from number-changing reactions. Rest-mass energy released by $3\to 2$ reactions heats the remaining particles, so the sector cools much more slowly than an isolated number-conserving nonrelativistic gas. The temperature laws follow directly from cosmological entropy conservation. An example is discussed in Cannibal Dark Matter.

If relic particle number and reference-gas entropy are separately conserved in a comoving volume, their ratio $Y=n/s$ is constant. A decay or entropy injection can change it. The reference entropy is that of the specified thermal sector, not necessarily the entropy of the decoupled relic itself.

stationary convection has a time-independent velocity and temperature field, in the chosen fixed frame. Its onset from a conduction state corresponds to a real growth rate passing through zero. Stability of a resulting steady convection roll is a separate nonlinear question.