* Sharkovskii's theorem

In mathematics, Sharkovskii's theorem, named after Oleksandr Mikolaiovich Sharkovsky, is a result about discrete dynamical systems.

Sharkovskii's theorem states that if f has a periodic point of least period m and m precedes n in the above ordering, then f has also a periodic point of least period n.

Sharkovskii's theorem does not state that there are stable cycles of those periods, just that there are cycles of those periods.

# REDIRECT Sharkovskii's theorem

# REDIRECT Sharkovskii's theorem

A spectacular application of the methods of symbolic dynamics is Sharkovskii's theorem about periodic orbits of a continuous map of an interval into itself ( 1964 ).

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 ).

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.

Using the above theorem it is easy to see that the original Borsuk Ulam statement is correct since if we take a map f: S < sup > n </ sup > → ℝ < sup > n </ sup > that does not equalize on any antipodes then we can construct a map g: S < sup > n </ sup > → S < sup > n-1 </ sup > by the formula

In these more general contexts, the bracket does not have the meaning of an inner product, because the Riesz representation theorem does not apply.

However, Bell's theorem does not apply to all possible philosophically realist theories.

The completeness theorem is a central property of first-order logic that does not hold for all logics.

Second-order logic, for example, does not have a completeness theorem for its standard semantics ( but does have the completeness property for Henkin semantics ), and the same is true of all higher-order logics.

If F is continuous then under the null hypothesis converges to the Kolmogorov distribution, which does not depend on F. This result may also be known as the Kolmogorov theorem ; see Kolmogorov's theorem for disambiguation.

The Nagata – Smirnov metrization theorem, described below, provides a more specific theorem where the converse does hold.

In interacting quantum field theories, Haag's theorem states that the interaction picture does not exist.

For instance, in Heyting arithmetic, one can prove that for any proposition p which does not contain quantifiers, is a theorem ( where x, y, z ... are the free variables in the proposition p ).

However, in certain axiom systems for constructive set theory, the axiom of choice does imply the law of the excluded middle ( in the presence of other axioms ), as shown by the Diaconescu-Goodman-Myhill theorem.

* The no-cloning theorem does not prevent superluminal communication via quantum entanglement, as cloning is a sufficient condition for such communication, but not a necessary one.

This notation ( and the theorem ) does not say anything about the limit of the difference of the two functions as x approaches infinity.

Teleportation does not result in the copying of qubits, and hence is consistent with the no cloning theorem.

It is important to note that Rice's theorem does not say anything about those properties of machines or programs which are not also properties of functions and languages.

The structured program theorem does not address how to write and analyze a usefully structured program.

Such a theorem does not state that B is always true, only that B must be true if A is true.

However, the virial theorem does not depend on the notion of temperature and holds even for systems that are not in thermal equilibrium.

The constructive proof of the theorem leads to an understanding of the aliasing that can occur when a sampling system does not satisfy the conditions of the theorem.

By setting K = ker ( f ) we immediately get the first isomorphism theorem.

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.

A slight generalization of the theorem, which immediately follows from it, is: if p is prime and m and n are positive integers such that

or some variant immediately after a contradiction symbol ; this occurs in a proof by contradiction, to indicate that the original assumption was false and that the theorem must therefore be true.

If λI − T is invertible then that inverse is linear ( this follows immediately from the linearity of λI − T ), and by the bounded inverse theorem is bounded.

is invertible at a point p, then F is an invertible function near p. This follows immediately from the theorem above.

However, this way of stating a Fourier inversion theorem obscures several potential complications not immediately apparent.

If an F quadrupole and a D quadrupole are placed immediately next to each other, their fields completely cancel out ( in accordance with Earnshaw's theorem ).

More generally, the theorem follows immediately from the Crofton formula in integral geometry according to which the length of any curve equals the measure of the set of lines that cross the curve, multiplied by their numbers of crossings.

The transcendence of the following numbers follows immediately from the theorem:

Because this characterization is unaffected by graph complementation, it immediately implies the weak perfect graph theorem.

The theorem was first proved by Stephen Smale and is the fundamental result in the theory of high-dimensional manifolds: for a start, it almost immediately proves the Generalized Poincaré Conjecture.

Because Berge's forbidden graph characterization is self-complementary, the weak perfect graph theorem follows immediately from the strong perfect graph theorem.

As Paul Erdős observed, the Sylvester – Gallai theorem immediately implies that any set of n points that are not collinear determines at least n different lines.

Almost immediately, John Milnor observed that a theorem due to Ernst Witt implied the existence of a pair of 16-dimensional tori that have the same eigenvalues but different shapes.

The Lasker – Noether theorem follows immediately from the following three facts:

In this case, the Brouwer fixed point theorem follows almost immediately from the intermediate value theorem.

Proof: Follows immediately from Ptolemy's theorem:

Scarf, then at Stanford, had met me at the San Francisco Airport in December 1961, and as he was driving to Palo Alto on the freeway, one of us, in one sentence, provided a key to the solution ; the other, also in one sentence, immediately provided the other key ; and the lock clicked open .” This collaboration yielded their 1963 paper: `` A limit theorem on the core of an economy ,” which is one of the most fundamental results in general equilibrium theory.

The theorem almost immediately implies that if a finite group G can be realized as the Galois group of a Galois extension N of