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This theorem characterizes truth in an ultraproduct. Łoś's theorem says that a first-order formula holds in an ultraproduct exactly when it holds in a set of factors belonging to the ultrafilter.

The method of types combines two facts: there are few possible empirical distributions, and strings within each type class are equiprobable under an independent and identically distributed source. Information entropy determines how many strings share a frequency profile; relative entropy determines how unlikely that profile is under a specified source law. The connection is useful both for lossless source coding and for understanding rare observations.

The following formulas use natural logarithms, so information entropy and relative entropy are measured in nats. DefineHere $p$ is the source probability mass function and $q$ is an attainable empirical distribution. Use $0\ln 0=0$; first assume every $p(a)>0$. Base-two logarithms give the equivalent formulas with $2^{n{H}_2(q)}$ and $2^{-n{D}_2(q\Vert p)}$.

For a rational homology three-sphere $Y$, the linking form is the nonsingular pairingReversing the orientation of $Y$ negates this form.

Here the paired elements lie in the torsion subgroup of first homology.

For a sheet moving with $u(x,0)=\alpha x$ through otherwise stationary fluid, the layer thickness is $\delta =\sqrt{\nu /\alpha }$. Withthe boundary-layer equations reduce toThe exact solution is $f=1-e^{-\eta }$.

The twisted cubic is a nondegenerate projective variety, so every hypersurface containing it has degree at least two. At least two hypersurfaces are needed to cut out a curve in ${\mathbb{P}}^3$, making their degree product at least four rather than the curve's degree three.

A projective variety is nondegenerate when it is contained in no hyperplane of its ambient projective space.

Zeeman's theorem says that the complement of an $m$-twist spin fibers over the circle with fiber the punctured $m$-fold cyclic branched cover of the original knot. In particular, the $1$- and $-1$-twist spins are unknotted.

This identifies the complement structure of a twist-spun knot.

A closed-cover intersection statement for a simplex is equivalent to the nonexistence of a boundary-valued map that preserves every face.

Normalized distances to three closed sets give continuous barycentric coordinates; a covering forces at least one coordinate to vanish, placing the image on the simplex boundary.

An equilateral triangle has three equal side lengths.

An acute triangle has all three interior angles strictly smaller than a right angle.

An anti-trapped surface is the time reverse of a trapped surface: both future-directed orthogonal null expansions are positive.

A scalar conservation law in one space dimension has the form $u_t+{\partial }_{x}f(u)=0$. While a classical solution exists, it obeys the quasilinear transport equation $u_t+f^{\prime }(u)u_x=0$.

A linear transport equation has a prescribed velocity field and is linear in the transported unknown.

For the transport field $b(t,x)$, a characteristic curve solvesThe chain rule converts differentiation of $u(t,X(t))$ into the transport operator applied to $u$.

The Lorenz condition ${\partial }^{\mu }{\bar{h}}_{\mu \nu }=0$ reduces the vacuum field equation to $\square {\bar{h}}_{\mu \nu }=0$. A plane-wave amplitude therefore satisfies $k^{\mu }{H}_{\mu \nu }=0$ and $k^{\mu }{k}_{\mu }=0$.

For the conserved scalar stress tensor above,

A toric variety $X_{\Sigma }$ is obtained by gluing the affine toric varieties associated with the cones of a fan $\Sigma$. It contains an algebraic torus as a dense open orbit.

A fan in toric geometry is a finite collection of rational strongly convex cones in toric geometry in a real vector space with a lattice, closed under taking faces of a polyhedral cone, such that the intersection of two cones is a face of each. A fan determines a toric variety.

A fan $\Sigma$ in $N_{\mathbb{R}}$ is a finite collection of strictly convex rational polyhedral cones closed under taking faces, such that the intersection of two cones is a face of each.

An algebraic torus of dimension $n$ is an algebraic group isomorphic over the base field to $({\mathbb{G}}_m{)}^n$. Its character and cocharacter lattices are dual free abelian groups $M$ and $N$.