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At fixed cylindrical radius $R$ with $|z|\ll R$, the central gravitational potential is harmonic in height. The vertical epicyclic frequency equals ${\Omega }_K$ in this spherical potential.

Kinetic decoupling occurs when momentum-exchange collisions cease to maintain the temperature and velocity dispersion of a particle species in equilibrium with a surrounding bath. Afterwards, a freely moving nonrelativistic particle has momentum proportional to $a^{-1}$ and effective kinetic temperature proportional to $a^{-2}$, where $a$ is the scale factor.

Double-diffusive convection occurs when two properties, such as temperature and composition, affect buoyancy but diffuse at different rates. A stabilizing composition gradient can oppose a destabilizing thermal gradient and change the form of heat transport.

Convective overshoot carries material from an unstable convecting region into an adjacent stably stratified region because its momentum does not vanish at the stability boundary. In stars this mixes material beyond the boundary determined by a local stellar convective stability criterion, changing stellar core masses and nuclear lifetimes.

A stable stratification gives a restoring buoyancy force and positive squared buoyancy frequency for an adiabatically displaced parcel. An adverse specific entropy gradient can instead drive convection; other forces such as rotation can modify the full stability criterion.

A tensor-product operator factors as $A\otimes B$ on a tensor product of two spaces. If both factors are positive semidefinite operators, their tensor-product operator is positive: a tensor-product basis of eigenvectors has eigenvalues $a_i{b}_j\ge 0$. This positivity underlies the necessary direction of the positive partial transpose criterion.

For massless neutrinos with a temperature perturbation $\Theta$ and background energy density ${\bar{\rho }}_{\nu }$, the physical trace-free spatial stress-energy tensor isSome conventions define ${\Pi }^{ij}=-{\pi }_{\nu }^{ij}$ and correspondingly reverse the sign of the stress source in the gravitational-wave equation.

In natural units, a thermal photon gas with two polarizations has

Inverse temperature is $\beta =1/(k_{B}T)$, where $k_B$ is the Boltzmann constant and $T$ is the absolute temperature. A canonical ensemble weights an energy $E$ by its Boltzmann factor $e^{-\beta E}$. Calling couplings dimensionless means absorbing this factor $\beta$ into their energy coefficients. The alternative convention $k_B=1$ makes inverse temperature $1/T$; physical and dimensionless couplings must not be mixed.

In a thermal Euclidean worldline path integral with $\hslash =1$, $\beta$ is also the circumference of the imaginary-time circle.

A closed transversal is a closed curve everywhere transverse to the leaves of a foliation. A single such curve meeting every leaf proves that the foliation is a taut foliation.

An anti-symplectic map between symplectic manifolds is a diffeomorphism $f$ satisfying $f^{\ast }{\omega }^{\prime }=-\omega$. For symplectic vector spaces, the analogous definition uses an invertible linear map.

A Hamiltonian group action is a Lie group action on a symplectic manifold with an equivariant moment map $\mu :M\to {\mathfrak{g}}^{\ast }$. In the sign convention ${\iota }_{X_H}\omega =dH$, each component satisfies $d⟨\mu ,X⟩={\iota }_{X_M}\omega$, where $X_M$ is the fundamental vector field. The opposite Hamiltonian vector field convention reverses the sign.

A real symmetric matrix $A$ is copositive if its quadratic form satisfies $x^{T}Ax\ge 0$ for every $x$ in the nonnegative orthant. Every positive semidefinite matrix and every symmetric nonnegative matrix is copositive, as is their sum. The converse fails for the Horn copositive matrix.

The symmetric part of a real square matrix is $H=(A+A^T)/2$. It controls the quadratic form because $x^{T}Ax=x^{T}Hx$; the antisymmetric part contributes zero. In a linear differential equation $x^{\prime }=Ax$, it determines the instantaneous derivative of the squared Euclidean norm.

The two-sided full shift on a finite alphabet $[k]$ consists of all functions $z:\mathbb{Z}\to [k]$, with the product topology and the left shift. It is a compact metric space. A compatible metric isAgreement on increasingly large finite coordinate sets is equivalent to convergence in this topology. The metric above is compatible but is not invariant under the left shift.

A two-sided sequence over a finite alphabet is uniformly recurrent if every finite word over an alphabet occurring in it occurs with bounded gaps. More precisely, for each such word some $M$ ensures that every length-$M$ interval contains a complete occurrence. This is equivalent to being a minimal point of the full shift: a finite cover of the orbit closure by preimages of a word's cylinder set bounds its return gaps; conversely, bounded gaps pass to all points of the orbit closure and make every forward orbit dense there. The property concerns finite words, not infinite integer intervals.

The restricted mean survival time to a fixed cutoff $\tau$ is $\mathbb{E}[\min (T,\tau )]={\int }_0^{\tau }S(t)dt$ for a nonnegative event time. This follows by the tail integral formula for moments. Unlike the unrestricted mean, it uses no tail beyond $\tau$; under independent censoring it can be estimated by integrating a Kaplan–Meier estimator over adequately observed follow-up.

For a real symmetric matrix $X$ with eigenvalues ${\lambda }_1\ge \cdots \ge {\lambda }_n$, the sum of its $k$ largest eigenvalues is the support function of the fantope:Here $⪯$ is the Loewner order. In an orthonormal eigenbasis for $X$, the objective depends only on the diagonal entries of $Y$, which lie in the capped simplex. Thus the identity reduces to the sum of the largest components of the eigenvalues.

If $x\notin \mathrm{range}{P}^T=(\ker P{)}^{\perp }$, the support function is infinite along a kernel line. Otherwise linear programming duality gives the displayed minimum, attained because the corresponding linear programs are feasible with finite values. Splitting $q=\alpha -\beta$ into nonnegative parts yields the equivalent minimum of $1^T(\alpha +\beta )$. Thus worst-case revenue over a polyhedral uncertainty set is $r_0^{T}x-{\sigma }_C(x)$, with value $-\infty$ off the row space.

Superspace integration uses the Berezin integral over Grassmann variables. A chiral $d^2\theta$ integral extracts an F-term, while a full $d^4\theta$ integral extracts a D-term.