Let $c_s^2=(dP/d\Sigma {)}_0$ be the barotropic closure of a razor-thin disk sound-speed squared and ${\kappa }_r$ the radial epicyclic frequency, with ${\kappa }_r^2=2(2-q_r){\Omega }^2$ for orbital shear parameter $q_r$. For a radial axisymmetric normal mode in a uniform barotropic shearing sheet, let $U_x,U_y$ be velocity amplitudes and $\sigma$ the surface-density amplitude. Linear momentum gives $-i\omega U_x=2\Omega U_y-ik{c}_s^2\sigma /{\Sigma }_0$ and $-i\omega U_y=-{\kappa }_r^2{U}_x/(2\Omega )$; mass conservation gives $\omega \sigma =k{\Sigma }_0{U}_x$. Eliminating the amplitudes for $\omega \ne 0$ proves the displayed dispersion relation. The rotational term is epicyclic restoration; the pressure term is acoustic restoration. Without a perturbed gravitational potential, no self-gravity term belongs in the formula.

An increasing chain of countable sets has union of cardinality at most ${\aleph }_1$. If the union is uncountable, select ${\aleph }_1$ of its points and one containing member for each. Every countable member misses one selected point, so linear comparison puts it below that point's chosen member. The chosen subfamily is cofinal, and its union has size at most ${\aleph }_1$ by infinite cardinal arithmetic.

A properly embedded topological surface is boundary-parallel if it cobounds a product region with a topological surface in the ambient boundary, with the remaining boundary of that product contained in the ambient boundary. An annulus parallel into one boundary component has zero net oriented boundary class on that component.

If an infinite computably enumerable family contained only incompressible strings, select its first enumerated string of length at least $n$. This is a total computable selection rule, and its selected string has a description of length $O(\log n)$ encoding the threshold. For large $n$ that is shorter than the selected string. This contradiction proves immunity.

If an inclusion-maximal separated set has separation at least $r$, every point of the ambient metric space is at distance strictly less than $r$ from a selected point. Otherwise that point could be added. Thus upper bounds on the sizes or measures of radius-$r$ balls give lower bounds on the size of the separated set.

The state annihilated by every annihilation operator of the positive-frequency mode basis chosen in the asymptotic past. A Bogoliubov transformation can make it contain particles when tested by future out-mode number operators.

The state annihilated by every annihilation operator of the positive-frequency mode basis chosen in the asymptotic future. It need not coincide with the in-vacuum on a time-dependent spacetime.

A logarithmic divergence grows like a logarithm as a cutoff is removed. For example ${\int }_{ϵ}^{1}dx/x=\log (1/ϵ)\to \infty$ as $ϵ\to 0^{+}$. An integrand $f(x)=C/x+O(1)$ near zero gives $C\log (1/ϵ)+O(1)$. This behavior distinguishes logarithmic sensitivity to an infrared or ultraviolet cutoff from power-law divergence.

An idele is an element of the idele group: a tuple of nonzero elements of the completion of a valued field at all places, with finite-place components in the local unit groups except at finitely many places.

The idelic modulus is the product of normalized local moduli: ordinary real modulus, squared complex modulus and $(N\mathfrak{p}{)}^{-{\mathrm{ord}}_{\mathfrak{p}}(x)}$ at finite places. This continuous homomorphism to ${\mathbb{R}}_{>0}$ has the diagonal multiplicative group in its kernel by the product formula. It differs from the idele norm, which maps between idele groups of field extensions.

The map from an idele to its finite-place fractional ideal is surjective, has kernel the product of the infinite multiplicative groups and finite-place unit groups, and sends diagonal elements to principal fractional ideals. It realizes the ideal class group as the corresponding quotient of the idele group.

For a $y$-independent solenoidal magnetic field, the transverse components can be written $B_x=-{\psi }_z$ and $B_z={\psi }_x$. Contours of $\psi$ are transverse magnetic field lines. The independent axial component $B_y$ can twist the flux tube. Under the MHD induction equation, an additive time-dependent gauge can be chosen so that $D\psi /Dt=0$.

For $p>0$, the relative potential $GM/(a^p+r^p{)}^{1/p}$ generates density $(p+1)M{a}^p{r}^{p-2}/[4\pi (a^p+r^p{)}^{2+1/p}]$. In unit scales the separable augmented density is $(p+1)r^{p-2}{\psi }^{2p+1}/(4\pi )$. Its Jeans moments from an augmented density give constant anisotropy $\beta =1-p/2$.

In unit scales $I=\int \rho \psi d^{3}x=(p+1)\Gamma (1+1/p{)}^2/[p\Gamma (2+2/p)]$. The local virial relation of the hypervirial model gives global energies $T=I/4$ and $W=-I/2$. Restoring physical scales multiplies each integral by $G{M}^2/a$.

A container fingerprint is a small vertex subset $T$, usually contained in a hypergraph independent set $I$, which determines a container $C(T)$ containing $I$. A deterministic membership-query algorithm records positive answers in $T$ and uses negative answers to remove vertices from the container. Reconstruction from $T$ follows the golden rule for container algorithms.

A container algorithm may use membership in the unknown independent set only through information recorded in its container fingerprint. With a deterministic ordering and deterministic tests, record every positive queried vertex; queried negative vertices can be excluded. All state updates must be recoverable from the fingerprint alone, so replaying the algorithm constructs a container without knowing the independent set.

For fixed uniformity, sufficiently small nonsingleton hypergraph codegrees relative to $d{\tau }^{|\sigma |-1}$ permit a family of containers whose degree measures are bounded below one by a fixed amount. Each container has a container fingerprint of size $O(\tau n)$. Iteration gives containers inducing at most an arbitrary fixed proportion of all hyperedges, with logarithmic family size $O(n\tau \log (1/\tau ))$. The constants depend on the uniformity and desired edge proportion, and the average degree must be positive.

For a simply connected space, if lower homotopy groups lie in an admissible Serre class, the Hurewicz map in the next degree is an isomorphism modulo that class, and the lower homology groups lie in the class. This applies to finitely generated groups and to finite groups supported at prescribed primes.

If $X$ and $A$ are simply connected and the pair is $(r-1)$-connected, with $r\ge 3$, its first possible nonzero relative homotopy group maps isomorphically to relative integral homology. This lets a basis of a free relative homology group be realized by attaching disks.

A Huber loss applied to a discrete gradient combines quadratic penalization of small slopes with linear penalization of large slopes. This interpolates between squared-gradient smoothing and total variation denoising, allowing smooth regions while retaining an edge-preserving large-gradient regime. The threshold should be chosen consistently with the signal's units and discretization. It can reduce staircasing in total variation denoising without guaranteeing its elimination.