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.

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.

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.

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.

* 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.

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,

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

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

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

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

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.

In the primary decomposition theorem, it is not necessary that the vector space V be finite dimensional, nor is it necessary for parts ( A ) and ( B ) that P be the minimal polynomial for T.

One motivation for this use is that a number of generally accepted mathematical results, such as Tychonoff's theorem, require the axiom of choice for their proofs.

The debate is interesting enough, however, that it is considered of note when a theorem in ZFC ( ZF plus AC ) is logically equivalent ( with just the ZF axioms ) to the axiom of choice, and mathematicians look for results that require the axiom of choice to be false, though this type of deduction is less common than the type which requires the axiom of choice to be true.

Assuming ZF is consistent, Paul Cohen employed the technique of forcing, developed for this purpose, to show that the axiom of choice itself is not a theorem of ZF by constructing a much more complex model which satisfies ZF ¬ C ( ZF with the negation of AC added as axiom ) and thus showing that ZF ¬ C is consistent.

** Stone's representation theorem for Boolean algebras needs the Boolean prime ideal theorem.

** Gödel's completeness theorem for first-order logic: every consistent set of first-order sentences has a completion.

The original Nernst heat theorem makes the weaker and less controversial claim that the entropy change for any isothermal process approaches zero as T → 0:

This explains the failure of the classical equipartition theorem for metals that eluded classical physicists in the late 19th century.

Mordell's theorem had an ad hoc proof ; Weil began the separation of the infinite descent argument into two types of structural approach, by means of height functions for sizing rational points, and by means of Galois cohomology, which was not to be clearly named as that for two more decades.

The " heuristic " approach of the Logic Theory Machine tried to emulate human mathematicians, and could not guarantee that a proof could be found for every valid theorem even in principle.

* Metamath-a language for developing strictly formalized mathematical definitions and proofs accompanied by a proof checker for this language and a growing database of thousands of proved theorems ; while the Metamath language is not accompanied with an automated theorem prover, it can be regarded as important because the formal language behind it allows development of such a software ; as of March, 2012, there is no " widely " known such software, so it is not a subject of " automated theorem proving " ( it can become such a subject ), but it is a proof assistant.

Bandwidth typically refers to baseband bandwidth in the context of, for example, sampling theorem and Nyquist sampling rate, while it refers to passband bandwidth in the context of Nyquist symbol rate or Shannon-Hartley channel capacity for communication systems.

The binomial theorem also holds for two commuting elements of a Banach algebra.

Given x ∈ A, the holomorphic functional calculus allows to define ƒ ( x ) ∈ A for any function ƒ holomorphic in a neighborhood of Furthermore, the spectral mapping theorem holds:

Greek mathematician Euclid mentioned the special case of the binomial theorem for exponent 2 as did the 3rd century B. C.

The most basic example of the binomial theorem is the formula for the square of x + y:

If one integrates this picture, which corresponds to applying the fundamental theorem of calculus, one obtains Cavalieri's quadrature formula, the integral – see proof of Cavalieri's quadrature formula for details.