Liouville's theorem states that any bounded entire function must be constant.

Liouville's theorem may be used to elegantly prove the fundamental theorem of algebra.

As a consequence of Liouville's theorem, any function that is entire on the whole Riemann sphere ( complex plane and the point at infinity ) is constant.

( this is Liouville's theorem ).

Cantor points out that his constructions prove more — namely, they provide a new proof of Liouville's theorem: Every interval contains infinitely many transcendental numbers.

By Liouville's theorem, Hamiltonian flows preserve the volume form on the phase space.

The postulate is justified in part, for classical systems, by Liouville's theorem ( Hamiltonian ), which shows that if the distribution of system points through accessible phase space is uniform at some time, it remains so at later times.

A classical theorem of Joseph Liouville called Liouville's theorem shows the higher-dimensions have less varied conformal maps:

Liouville's theorem states this measure is invariant under the Hamiltonian flow.

* Liouville's theorem ( Hamiltonian )

According to Liouville's theorem, any bounded complex analytic function defined on the whole complex plane is constant.

Such a doubly periodic function cannot be holomorphic, as it would then be a bounded entire function, and by Liouville's theorem every such must be constant.

* In complex analysis, see Liouville's theorem ( complex analysis ); there is also a related theorem on harmonic functions.

* In conformal mappings, see Liouville's theorem ( conformal mappings ).

* In Hamiltonian mechanics, see Liouville's theorem ( Hamiltonian ).

* In differential algebra, see Liouville's theorem ( differential algebra )

Liouville's theorem implies invariance of a kinematic measure on the unit tangent bundle.

* Liouville's theorem ( Hamiltonian )

By Liouville's theorem, each symplectomorphism preserves the volume form on the phase space.

This is called Liouville's theorem.

By the vector-valued version of Liouville's theorem, this function is constant, thus everywhere zero as it is zero at infinity.

Not all closed-form expressions have closed-form antiderivatives ; this study forms the subject of differential Galois theory, which was initially developed by Joseph Liouville in the 1830s and 1840s, leading to Liouville's theorem which classifies which expressions have closed form antiderivatives.

Consider the Laurent expansion at all such z and subtract off the singular part: we are left with a function on the Riemann sphere with values in C, which by Liouville's theorem is constant.

The relationship between the microcanonical ensemble, Liouville's theorem, and ergodic hypothesis can be summarized as follows: The key assumption of a microcanonical ensemble is that all accessible microstates are equally probable.

For instance, all Hamiltonian systems are volume-preserving because of Liouville's theorem.

In mathematical physics, Liouville made two fundamental contributions: the Sturm – Liouville theory, which was joint work with Charles François Sturm, and is now a standard procedure to solve certain types of integral equations by developing into eigenfunctions, and the fact ( also known as Liouville's theorem ) that time evolution is measure preserving for a Hamiltonian system.

In complex analysis, Liouville's theorem, named after Joseph Liouville, states that every bounded entire function must be constant.

In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics.

The basic theorem of differential Galois theory is due to Joseph Liouville in the 1830s and 1840s and hence referred to as Liouville's theorem.

This reduction has been accomplished by the general methods of linear algebra, i.e., by the primary decomposition theorem.

The latter theorem has been generalized by Yamabe and Yujobo, and Cairns to show that in Af there are families of such cubes.

* Every continuous functor on a small-complete category which satisfies the appropriate solution set condition has a left-adjoint ( the Freyd adjoint functor theorem ).

** The theorem that every Hilbert space has an orthonormal basis.

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

Note that " completeness " has a different meaning here than it does in the context of Gödel's first incompleteness theorem, which states that no recursive, consistent set of non-logical axioms of the Theory of Arithmetic is complete, in the sense that there will always exist an arithmetic statement such that neither nor can be proved from the given set of axioms.

has no zero in F. By contrast, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed.

The no-hair theorem states that, once it achieves a stable condition after formation, a black hole has only three independent physical properties: mass, charge, and angular momentum.

The use of Bayesian probabilities as the basis of Bayesian inference has been supported by several arguments, such as the Cox axioms, the Dutch book argument, arguments based on decision theory and de Finetti's theorem.

It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem of calculus.

* Duality: Every statement, theorem, or definition in category theory has a dual which is essentially obtained by " reversing all the arrows ".

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.

# X has a sub-base such that every cover of the space by members of the sub-base has a finite subcover ( Alexander's sub-base theorem )

In this situation, the chain rule represents the fact that the derivative of is the composite of the derivative of f and the derivative of g. This theorem is an immediate consequence of the higher dimensional chain rule given above, and it has exactly the same formula.

In his 1799 doctorate in absentia, A new proof of the theorem that every integral rational algebraic function of one variable can be resolved into real factors of the first or second degree, Gauss proved the fundamental theorem of algebra which states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.

Moreover, the Banach fixed point theorem states that every contraction mapping on a nonempty complete metric space has a unique fixed point, and that for any x in M the iterated function sequence x, f ( x ), f ( f ( x )), f ( f ( f ( x ))), ... converges to the fixed point.

* Every pair of congruence relations for an unknown integer x, of the form x ≡ k ( mod a ) and x ≡ l ( mod b ), has a solution, as stated by the Chinese remainder theorem ; in fact the solutions are described by a single congruence relation modulo ab.

The classification theorem has applications in many branches of mathematics, as questions about the structure of finite groups ( and their action on other mathematical objects ) can sometimes be reduced to questions about finite simple groups.

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.

It follows from this theorem that all Carmichael numbers are odd, since any even composite number that is square-free ( and hence has only one prime factor of two ) will have at least one odd prime factor, and thus results in an even dividing an odd, a contradiction.

Formal verification of statements in logic has been necessary for software development of safety-critical systems, and advances in automated theorem proving have been driven by this need.

The study of mathematical proof is particularly important in logic, and has applications to automated theorem proving and formal verification of software.

For systems where the volume is preserved by the flow, Poincaré discovered the recurrence theorem: Assume the phase space has a finite Liouville volume and let F be a phase space volume-preserving map and A a subset of the phase space.