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A set is hereditarily countable when its transitive closure is countable. These sets form the transitive set $H_{{\omega }_1}$ and all have rank below ${\omega }_1$.

The relativization ${\phi }^M$ of a formula $\phi$ is obtained recursively by restricting every quantifier to $M$:For parameters in $M$, the ambient statement ${\phi }^M$ holds exactly when the structure $(M,\in )$ satisfies $\phi$.

If a transitive class $M$ contains the empty set and is closed under pairing and union, then $(M,\in )$ satisfies extensionality, empty set, pairing, and union. Transitivity makes all members of each $x\in M$ visible inside $M$, so the ambient witnesses have the same required membership relations internally.

For nonzero $x\in K$, choose $y=x^{-1}$. Thenso the trace form is nondegenerate.

The positive semidefinite trace ball of radius $s$ isIt is closed and convex, and $\Vert Z{\Vert }_F\le \mathrm{tr}Z\le s$ for every $Z\in {\mathcal{S}}_s$.

An initial segment contains every element below each of its elements.

An element is greatest when every element of the ordered set is less than or equal to it.

The empty order is the unique total order on the empty set.

A well-order is a total order in which every nonempty subset has a least element.

For the standard longitude $a$ and meridian $b$,

The Cox construction presents a toric variety as a quotient of an open subset of affine space by a quasitorus determined by its divisor class group.

Each ray $\rho$ of a fan determines a torus-invariant prime divisor $D_{\rho }$. Integer combinations of these divisors encode line bundles and maps from a toric variety.

A lattice homomorphism that maps each cone of one fan into a cone of another induces an equivariant morphism of the corresponding toric varieties.

The Hirzebruch surface is the ruled surfaceIt is the smooth toric surface with rays $(1,0),(0,1),(-1,k),(0,-1)$.

This is a two-dimensional weighted projective space. The weighted projective plane $\mathbb{P}(a,b,c)$ is the quotient of ${\mathbb{A}}^3\setminus \{0\}$ by $\lambda \cdot (x,y,z)=({\lambda }^{a}x,{\lambda }^{b}y,{\lambda }^{c}z)$. It is generally singular but is a proper toric surface.

An affine toric chart associated with a cone $\sigma$ is smooth exactly when the primitive ray generators of $\sigma$ extend to a lattice basis. For a two-dimensional cone generated by $u,v$, this is equivalent to $|\det (u,v)|=1$.

A toric variety is proper exactly when its fan is complete.

The orbit-cone correspondence assigns to every cone $\sigma \in \Sigma$ a torus orbit of codimension $\dim \sigma$. In particular, maximal cones correspond to torus-fixed points.

For baryon-loading ratio $R=3{\rho }_b/(4{\rho }_{\gamma })$, the photon-baryon sound speed is $c_s^2=1/[3(1+R)]$.

Tidal stripping removes satellite material outside the tidal radius. The resulting mass loss weakens dynamical friction and can greatly delay or prevent a merger.