A pointclass is said to have the uniformization property if every relation in can be uniformized by a partial function in.

It follows from ZFC alone that and have the uniformization property.

* and have the uniformization property for every natural number.

* Therefore, the collection of projective sets has the uniformization property.

In fact, L ( R ) does not have the uniformization property ( equivalently, L ( R ) does not satisfy the axiom of uniformization ).

The geometrization conjecture is an analogue for 3-manifolds of the uniformization theorem for surfaces.

The uniformization theorem for surfaces states that the upper half-plane is the universal covering space of surfaces with constant negative Gaussian curvature.

In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of the three domains: the open unit disk, the complex plane, or the Riemann sphere.

The uniformization theorem is a generalization of the Riemann mapping theorem from proper simply connected open subsets of the plane to arbitrary simply connected Riemann surfaces.

that it is nevertheless possible to prove the uniformization theorem via Ricci flow.

Koebe proved the general uniformization theorem that if a Riemann surface is homeomorphic to an open subset of the complex sphere ( or equivalently if every Jordan curve separates it ), then it is conformally equivalent to an open subset of the complex sphere.

The simultaneous uniformization theorem of Bers shows that it is possible to simultaneously uniformize two compact Riemann surfaces of the same genus > 1 with the same quasi-Fuchsian group.

) Indeed, a triumph of nineteenth century geometry was the proof of the uniformization theorem, the analogous topological classification of smooth two-manifolds, where Hamilton showed that the Ricci flow does indeed evolve a negatively curved two-manifold into a two-dimensional multi-holed torus which is locally isometric to the hyperbolic plane.

Low-dimensional topology is strongly geometric, as reflected in the uniformization theorem in 2 dimensions – every surface admits a constant curvature metric ; geometrically, it has one of 3 possible geometries: positive curvature / spherical, zero curvature / flat, negative curvature / hyperbolic – and the geometrization conjecture ( now theorem ) in 3 dimensions – every 3-manifold can be cut into pieces, each of which has one of 8 possible geometries.

2-dimensional topology can be studied as complex geometry in one variable ( Riemann surfaces are complex curves ) – by the uniformization theorem every conformal class of metrics is equivalent to a unique complex one, and 4-dimensional topology can be studied from the point of view of complex geometry in two variables ( complex surfaces ), though not every 4-manifold admits a complex structure.

By the uniformization theorem, any hyperbolic surface X – i. e., the Gaussian curvature of X is equal to negative one at every point – is covered by the hyperbolic plane.

AD < sub > R </ sub > is equivalent to AD plus the axiom of uniformization.

In mathematics, a Fuchsian model is a construction of a hyperbolic Riemann surface R as a quotient of the upper half-plane H. By the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic.

In set theory, the axiom of uniformization, a weak form of the axiom of choice, states that if is a subset of, where and are Polish spaces,

Such a function is called a uniformizing function for, or a uniformization of.

A uniformization of then picks exactly one element from each such subset, whenever the subset is nonempty.

The first rigorous proofs of the general uniformization theorem were given by and.

It entails the uniformization of analytic relations by means of automorphic functions.

The uniformization theorem implies a similar result for arbitrary connected second countable surfaces: they can be given Riemannian metrics of constant curvature.

Felix and conjectured the uniformization theorem for ( the Riemann surfaces of ) algebraic curves.

* William Goldman, " Conformally flat manifolds with nilpotent holonomy and the uniformization problem for 3-manifolds ", Transactions of the American Mathematical Society 278 ( 1983 ), 573 -- 583.

In 1911, the name changed to National Republican Guard: this was to be a security force consisting of military personnel organised in a special corps of troops depending, in peace time, on the Ministry of Internal Administration, for the purpose of conscription, administration and execution with regards to its mission, and the Ministry of the National Defense for the purpose of uniformization and normalization of the military doctrine, as well as for its armament and equipment.

Other choice axioms weaker than axiom of choice include the Boolean prime ideal theorem and the axiom of uniformization.

Also important are Plemelj's contributions to the theory of analytic functions in solving the problem of uniformization of algebraic functions, contributions on formulation of the theorem of analytic extension of designs and treatises in algebra and in number theory.

However, his proof relied on the uniformization theorem.

The measurable Riemann mapping theorem shows more generally that the map to an open subset of the complex sphere in the uniformization theorem can be chosen to be a quasiconformal map with any given bounded measurable Beltrami coefficient.

Note that the term " uniformization " correctly suggests a kind of smoothing away of irregularities in the geometry, while the term " geometrization " correctly suggests placing a geometry on a smooth manifold.

In particular, many discussions of uniformization speak of complex curves rather than real two-manifolds.

The Ricci flow does not preserve volume, so to be more careful in applying the Ricci flow to uniformization and geometrization one needs to normalize the Ricci flow to obtain a flow which preserves volume.

When Yau was a graduate student, he started to generalize the uniformization theorem of Riemann surfaces to higher-dimensional complex Kähler manifolds.

* Fredkin's concept of the multiverse as a finite automaton with absolute space, time, and information might be isomorphic to a sheaf uniformization axiom.

His Ethics defines `` possessions as the property of the community, of which the individual is sovereign steward.

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The board is diminished in both respects, while it retains control over zoning, franchises, pier leases, sale, leasing and assignment of property, and other trusteeship functions.

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When suitable equipment is located by the SBA representative, the small business concern is contacted and advised on when, where, and how to bid on such property.

Each applicant is required to own or have sufficient interest in the property to be explored.

if the Government certifies that production may be possible from the property, the royalty obligation continues for the 10-year period usually specified in the contract or until the Government's contribution is repaid with interest.

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Intangible property is taxable wherever the owner has a place of abode the greater portion of the year.

Although a similar situs for tangible property is mentioned in the statute, this is cancelled out by the provision that definite kinds of property `` and all other tangible property '' situated or being in any town is taxable where the property is situated.