If X is the toric variety corresponding to the normal fan of P, then P defines an ample line bundle on X, and the Ehrhart polynomial of P coincides with the Hilbert polynomial of this line bundle.

A sigma model whose target space is a toric variety may be described by an abelian gauge theory with charged chiral multiplets.

In algebraic geometry, a toric variety or torus embedding is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whole variety.

For each cone σ its affine toric variety U < sub > σ </ sub > is the spectrum of the semigroup algebra of the dual cone.

The fan associated with a toric variety condenses some important data about the variety.

A toric variety is nonsingular if its cones of maximal dimension are generated by a basis of the lattice.

This implies that every toric variety has a resolution of singularities given by another toric variety, which can be constructed by subdividing the maximal cones into cones of nonsingular toric varieties.

The toric variety of the polytope is the toric variety of its fan.

This is different from a spherical lens in that one or both surfaces of the lens deliver a toric correction.

Discrete geometry has large overlap with convex geometry and computational geometry, and is closely related to subjects such as finite geometry, combinatorial optimization, digital geometry, discrete differential geometry, geometric graph theory, toric geometry, and combinatorial topology.

which is used in the construction of toric varieties.

If f is a linear map from N < sub > 1 </ sub > to N < sub > 2 </ sub > such that the image of every cone of Δ < sub > 1 </ sub > is contained in a cone of Δ < sub > 2 </ sub >, then f induces a morphism f < sub >*</ sub > between the corresponding toric varieties.

A variation of this construction is to take a rational polytope in the dual of N and take the toric variety of its polar set in N.

The approach of toric geometry is to write

The triangle is the toric base of the complex projective plane.

An O-ring, also known as a packing, or a toric joint, is a mechanical gasket in the shape of a torus ; it is a loop of elastomer with a disc-shaped cross-section, designed to be seated in a groove and compressed during assembly between two or more parts, creating a seal at the interface.

A different model of toric lenses is created by Alcon and may correct up to 3 diopters of astigmatism.

This applies when X is toric, meaning that it is acted upon by a complex torus, or at least locally toric.

The first disposable toric lenses were introduced in 2000 by Vistakon.

Mirror symmetry, in a class of models of toric varieties with zero first Chern class Calabi-Yau manifolds and positive first Chern class ( Fano varieties ) was proven by Kentaro Hori and Cumrun Vafa.

Formulas for the other coefficients are much harder to get ; Todd classes of toric varieties, the Riemann – Roch theorem as well as Fourier analysis have been used for this purpose.

In their construction the mirror partners appear as anticanonically embedded hypersurfaces or certain complete intersections in Fano toric varieties.

* Home page of D. A. Cox, with several lectures on toric varieties

Several condensed matter systems are being studied with potential applications to quantum computation, including experimental systems like quantum dots, SQUIDs, and theoretical models like the toric code and the quantum dimer model.

In those with keratoconus, toric contact lenses often enable patients to achieve better visual acuities than eyeglasses.

In order to achieve the most benefit from a toric lens, the surgeon must rotate the lens to be on axis with the patient ’ s astigmatism.

A toric contact lens has a different focusing power horizontally than it does vertically, and as a result can correct for astigmatism.

) Because a toric lens must have the proper orientation to correct for a person's astigmatism, a toric contact lens must have additional design characteristics to prevent the lens from rotating out of the ideal alignment.

Some toric contact lenses have marks or etchings that can assist the eye doctor in fitting the lens.

In this generalized context the original mirror symmetry, which relates pairs of toric Calabi-Yau manifolds, relates the moduli spaces of 2-dimensional abelian supersymmetric gauge theories when the sums of the electric charges of the matter are equal to zero.

The toric variety has a map to the polytope in the dual of N whose fibers are topological tori.

At the time, there were no officially recognized romanizations of character and mecha names, and a variety of different spellings were being used in the English-language fan community.

Most use film, video, clips and still images from a variety of sources: American and Japanese science fiction movies of the 1950s, old newsreel footage, more modern TV news and cable show clips, atomic bomb test footage, old civil defense films, silent movie footage, 1950s social etiquette films, industrial films, clips from their various SubGenius events, self-created footage, and fan clips.

While the term was originally restricted to stories in which male media characters were involved in an explicit sexual relationship as a primary plot element ( also known as " slash ", " m / m slash ", " menslash ", " altfic "), it is now often used to refer to any fan story containing a pairing between same-sex characters, although many fans distinguish the female-focused variety as a separate genre commonly referred to as femslash ( also known as " f / f slash ", " femmeslash ", " altfic " and " saffic ").

Pern fandom consists of a large variety of fan communities.

This non-centralized movement has given birth to science fiction fanzines ( and amateur press associations ), science fiction conventions, the Hugo Awards ( and various imitators / derivatives ), filk music, " fan funds " such as the Trans Atlantic Fan Fund, and a variety of other institutions, jargon and customs.

Hazing has been reported in a variety of social contexts, including sports teams, academic fraternities and sororities ( see fraternities and sororities ); high schools ( in the sense of secondary education ); college and universities ; groups, like competition teams, fan clubs, social groups ; secret societies and even certain service clubs ; competitive sports teams ; and armed forces.

Anime Expo also has a large variety of focus panels, workshops, and events, some of which are fan or industry sponsored.

There are three primary components to a blower door: ( 1 ) a calibrated, variable-speed fan, capable of inducing a range of airflows sufficient to pressurize and depressurize a variety of building sizes, ( 2 ) a pressure measurement instrument, called a manometer, to simultaneously measure the pressure differential induced across the face of the fan and across the building envelope, as a result of fan airflow, and ( 3 ) a mounting system, used to mount the fan in a building opening, such as a door or a window.

A variety of blower door airtightness metrics can be produced using the combination of building-to-outside pressure and fan airflow measurements.

For example, a variety is smooth if every cone in its fan can be generated by a subset of a basis for the free abelian group N.

There're also third-party programs that work on a variety of motherboards and allow wide customization of fan behaviour depending

Dangers and crew performed a wide variety of hits and fan favorites from the entire back catalog, though relatively little of the new jazz fusion material from At the Center was played.

Enzai has become popular and successful in Japan, birthing two OVAs, several Drama CD Collections, a novelization, an Official Fanbook, and a variety of fan collectibles ( pin badges, phone cards, satchels, etc .).