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uniformization and theorem
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.
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.
Felix and conjectured the uniformization theorem for ( the Riemann surfaces of ) algebraic curves.
The first rigorous proofs of the general uniformization theorem were given by and.
However, his proof relied on the uniformization theorem.
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.
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.
) 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.
* uniformization theorem
When Yau was a graduate student, he started to generalize the uniformization theorem of Riemann surfaces to higher-dimensional complex Kähler manifolds.
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.
In the context of compact Riemann surfaces X, via the Riemann uniformization theorem, this can be seen as a distinction between the surfaces of different topologies:
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.
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.

uniformization and for
* 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.
Such a function is called a uniformizing function for, or a uniformization of.
The uniformization property is implied by the scale property, at least for adequate pointclasses of a certain form.
* and have the uniformization property for every natural number.

uniformization and arbitrary
Thus, allowing arbitrary sets X and Y ( rather than just Polish spaces ) would make the axiom of uniformization equivalent to AC.

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

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

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

theorem and implies
In constructive set theory, however, Diaconescu's theorem shows that the axiom of choice implies the law of the excluded middle ( unlike in Martin-Löf type theory, where it does not ).
This implies, by the Bolzano – Weierstrass theorem, that any infinite sequence from the set has a subsequence that converges to a point in the set.
The first case was done by the Gorenstein – Walter theorem which showed that the only simple groups are isomorphic to L < sub > 2 </ sub >( q ) for q odd or A < sub > 7 </ sub >, the second and third cases were done by the Alperin – Brauer – Gorenstein theorem which implies that the only simple groups are isomorphic to L < sub > 3 </ sub >( q ) or U < sub > 3 </ sub >( q ) for q odd or M < sub > 11 </ sub >, and the last case was done by Lyons who showed that U < sub > 3 </ sub >( 4 ) is the only simple possibility.
This is because if a group has sectional 2-rank at least 5 then MacWilliams showed that its Sylow 2-subgroups are connected, and the balance theorem implies that any simple group with connected Sylow 2-subgroups is either of component type or characteristic 2 type.
The three classes are groups of GF ( 2 ) type ( classified mainly by Timmesfeld ), groups of " standard type " for some odd prime ( classified by the Gilman – Griess theorem and work by several others ), and groups of uniqueness type, where a result of Aschbacher implies that there are no simple groups.
For the special case, this implies that the length of a vector is preserved as well — this is just Parseval's theorem:
In fact, Cantor's method of proof of this theorem implies the existence of an " infinity of infinities ".
Together with soundness ( whose verification is easy ), this theorem implies that a formula is logically valid if and only if it is the conclusion of a formal deduction.
This is an immediate consequence of the completeness theorem, because only a finite number of axioms from Γ can be mentioned in a formal deduction of φ, and the soundness of the deduction system then implies φ is a logical consequence of this finite set.
The Paley – Wiener theorem immediately implies that if f is a nonzero distribution of compact support ( these include functions of compact support ), then its Fourier transform is never compactly supported.
Shannon's theorem also implies that no lossless compression scheme can compress all messages.
Bell's theorem implies, and it has been proven mathematically, that compatible measurements cannot show Bell-like correlations, and thus entanglement is a fundamentally non-classical phenomenon.
One might expect that by the Hahn-Banach theorem for bounded linear functionals, every bounded linear functional on C < sub > c </ sub >( X ) extends in exactly one way to a bounded linear functional on C < sub > 0 </ sub >( X ), the latter being the closure of C < sub > c </ sub >( X ) in the supremum norm, and that for this reason the first statement implies the second.
The Heine – Borel theorem implies that a Euclidean n-sphere is compact.
The integrability condition and Stokes ' theorem implies that the value of the line integral connecting two points is independent of the path.
If we choose the volume to be a ball of radius a around the source point, then Gauss ' divergence theorem implies that
Cox's theorem implies that any plausibility model that meets the
The Arzelà – Ascoli theorem implies that if is a uniformly bounded sequence of functions with bounded Lipschitz constant, then it has a convergent subsequence.
For instance, Bell's theorem implies that quantum mechanics cannot satisfy both local realism and counterfactual definiteness.
Noether's theorem implies that there is a conserved current associated with translations through space and time.
However, if the errors are not normally distributed, a central limit theorem often nonetheless implies that the parameter estimates will be approximately normally distributed so long as the sample is reasonably large.
In fact these uses of AC are essential: in 1950 Kelley proved that Tychonoff's theorem implies the axiom of choice.
To prove that Tychonoff's theorem in its general version implies the axiom of choice, we establish that every infinite cartesian product of non-empty sets is nonempty.
Liouville's theorem implies invariance of a kinematic measure on the unit tangent bundle.

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