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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.

If a real algebraic number $\alpha$ has degree $d\ge 3$, then for every $\epsilon >0$ there are only finitely many reduced fractions $p/q$ such thatThe Thue-Siegel auxiliary polynomial proof chooses two such approximations, constructs an auxiliary polynomial nonzero at their pair, and compares its denominator lower bound with the upper bound supplied by its high-order zero at $(\alpha ,\alpha )$.

Let $F(X,Y)=P(X)+YQ(X)$ with $P,Q\in \mathbb{Z}[X]$ linearly independent, ${\mathrm{deg}}_{X}F\le n$, and $H(F)\le C^n$. For every $\delta >0$, if a reduced rational $p/q$ has sufficiently large denominator in terms of $C$ and $\delta$, then $F(X,y)$ has multiplicity at most $\delta n+1$ at $p/q$, for every fixed real $y$.

Indeed, the nonzero Wronskianhas degree below $2n$ and height at most $C_0^n$. A zero of multiplicity $r$ of $F(X,y)$ forces a zero of multiplicity at least $r-1$ of $W$. By Gauss lemma for polynomials, $(qX-p{)}^{r-1}$ then divides $W$ in $\mathbb{Z}[X]$, so $q^{r-1}$ divides the leading coefficient of $W$ and is at most $C_0^n$.