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

The structured program theorem provides the theoretical basis of structured programming.

In mechanics, the virial theorem provides a general equation relating the average over time of the total kinetic energy,, of a stable system consisting of N particles, bound by potential forces, with that of the total potential energy,, where angle brackets represent the average over time of the enclosed quantity.

The sampling theorem provides a sufficient condition, but not a necessary one, for perfect reconstruction.

( Section 2. 7 provides a category-theoretic presentation of the theorem as a colimit in the category of groupoids ).

The Nyquist-Shannon sampling theorem provides an important guideline as to how much digital data is needed to accurately portray a given analog signal.

Carlson's theorem provides necessary and sufficient conditions for a Newton series to be unique, if it exists.

The fundamental theorem of Galois theory provides a link between algebraic field extensions and group theory.

In mathematics, the Banach fixed-point theorem ( also known as the contraction mapping theorem or contraction mapping principle ) is an important tool in the theory of metric spaces ; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, and provides a constructive method to find those fixed points.

For example, Euclid provides an elaborate proof of the Pythagorean theorem, by using addition of areas instead of the much simpler proof from similar triangles, which relies on ratios of line segments.

) Thus, the group of conformal transformations in spaces of dimension greater than 2 are much more restricted than the planar case, where the Riemann mapping theorem provides a large group of conformal transformations.

In practice, we might not have the distribution function but the Fisher – Tippett – Gnedenko theorem provides the following asymptotic result

However, the choice of dimensionless parameters is not unique: Buckingham's theorem only provides a way of generating sets of dimensionless parameters, and will not choose the most ' physically meaningful '.

The Vaschy – Buckingham π theorem provides a method for computing sets of dimensionless parameters from the given variables, even if the form of the equation is still unknown.

However, the choice of dimensionless parameters is not unique: Buckingham's theorem only provides a way of generating sets of dimensionless parameters, and will not choose the most ' physically meaningful '.

According to Allen Weiss, in Mirrors of Infinity, this optical effect is a result of the use of the tenth theorem of Euclid ’ s Optics which asserts that “ the most distant parts of planes situated below the eye appear to be the most elevated .” In Fouquet ’ s time, interested parties could cross the canal in a boat, but walking around the canal provides a view of the woods that mark what is no longer the garden and shows the distortion of the grottos previously seen as sculptural.

In broad terms the spectral theorem provides conditions under which an operator or a matrix can be diagonalized ( that is, represented as a diagonal matrix in some basis ).

The spectral theorem also provides a canonical decomposition, called the spectral decomposition, eigenvalue decomposition, or eigendecomposition, of the underlying vector space on which the operator acts.

Fisher's factorization theorem or factorization criterion provides a convenient characterization of a sufficient statistic.

This theorem is an important tool in model theory, as it provides a useful method for constructing models of any set of sentences that is finitely consistent.

The Cayley – Hamilton theorem always provides a relationship between the powers of A ( though not always the simplest one ), which allows one to simplify expressions involving such powers, and evaluate them without having to compute the power A < sup > n </ sup > or any higher powers of A.

While this provides a valid proof ( for matrices over the complex numbers ), the argument is not very satisfactory, since the identities represented by the theorem do not in any way depend on the nature of the matrix ( diagonalizable or not ), nor on the kind of entries allowed ( for matrices with real entries the diagonizable ones do not form a dense set, and it seems strange one would have to consider complex matrices to see that the Cayley – Hamilton theorem holds for them ).

While we cannot do this by simply rearranging the quantifiers, we show that it is yet enough to prove the theorem for sentences of that form.

A theorem might be simple to state and yet be deep.

Solomonoff's universal prior probability of any prefix p of a computable sequence x is the sum of the probabilities of all programs ( for a universal computer ) that compute something starting with p. Given some p and any computable but unknown probability distribution from which x is sampled, the universal prior and Bayes ' theorem can be used to predict the yet unseen parts of x in optimal fashion.

* 1854 – Clausius establishes the importance of dQ / T ( Clausius's theorem ), but does not yet name the quantity.

Some philosophers believe that the " no-free-lunch in search and optimization theorem " of David Wolpert and William G. Macready is a probability-based extension of induction, yet this is misleading, as inductive logic accustomed to probabilistic arguments and the No free lunch theorem ( NFL ) is more a variation of economic rational choice theory.

The proof of a mathematical theorem exhibits mathematical elegance if it is surprisingly simple yet effective and constructive ; similarly, a computer program or algorithm is elegant if it uses a small amount of code to great effect.

Perfection of the complements of line graphs of perfect graphs is yet another restatement of König's theorem, and perfection of the line graphs themselves is a restatement of an earlier theorem of König, that every bipartite graph has an edge coloring using a number of colors equal to its maximum degree.

Using the close correspondence between divisors and holomorphic line bundles on a Riemann surface, the theorem can also be stated in a different, yet equivalent way: let L be a holomorphic line bundle on X.

The axioms I1, I2, and I3 were at first suspected to be inconsistent ( in ZFC ) as it was thought possible that Kunen's inconsistency theorem that Reinhardt cardinals are inconsistent with the axiom of choice could be extended to them, but this has not yet happened and they are now usually believed to be consistent.

Many consequences of the Löwenheim – Skolem theorem seemed counterintuitive to logicians in the early 20th century, as the distinction between first-order and non-first-order properties was not yet understood.

The universal prior probability of any prefix p of a computable sequence x is the sum of the probabilities of all programs ( for a universal computer ) that compute something starting with p. Given some p and any computable but unknown probability distribution from which x is sampled, the universal prior and Bayes ' theorem can be used to predict the yet unseen parts of x in optimal fashion.

Spencer-Brown's claimed proof of the four-color theorem has yet to find any defenders ; Kauffman provides a detailed review of parts of that work.

In complex analysis, a discipline within mathematics, the Picard theorem, named after Charles Émile Picard, is either of two distinct yet related theorems, both of which pertain to the range of an analytic function.

At the time, Gödel's completeness theorem and the compactness theorem had not yet been proved.

The ancient geometers are not done yet, for if the fifth vertex of the pentagon is marked as E and FE and BF are joined ( with FE = BF = z ), then cyclic quadrilateral EFBA will be formed with diagonals length d ( diameter ) and b. Applying the ' Almagest ' theorem yet again:

) In second order logic, however, the well-ordering theorem is strictly stronger than the axiom of choice: from the well-ordering theorem one may deduce the axiom of choice, but from the axiom of choice one cannot deduce the well-ordering theorem.

A stronger form of the twin prime conjecture, the Hardy – Littlewood conjecture, postulates a distribution law for twin primes akin to the prime number theorem.

Although this at first appears to be a stronger statement, it is a direct consequence of the other form of the theorem, through the use of successive polynomial division by linear factors.

However, a slightly stronger form of the theorem is true, and is known as Lehmer's theorem.

The stronger version of the incompleteness theorem that only assumes consistency, rather than ω-consistency, is now commonly known as Gödel's incompleteness theorem and as the Gödel – Rosser theorem.

The later version of this theorem is stronger — has weaker conditions — since monotonicity, non-imposition, and independence of irrelevant alternatives together imply Pareto efficiency, whereas Pareto efficiency, non-imposition, and independence of irrelevant alternatives together do not imply monotonicity.

In mathematical logic, Goodstein's theorem is a statement about the natural numbers, proved by Reuben Goodstein in 1944, which states that every Goodstein sequence eventually terminates at 0. showed that it is unprovable in Peano arithmetic ( but it can be proven in stronger systems, such as second order arithmetic ).

Compared to the second recursion theorem, the first recursion theorem produces a stronger conclusion but only when narrower hypotheses are satisfied.

An Euler probable prime to base a is an integer that is indicated prime by the somewhat stronger theorem that for any prime p, a < sup >( p − 1 )/ 2 </ sup > equals modulo p, where is the Legendre symbol.

One of his results was a 1901 theorem proving that the Riemann hypothesis is equivalent to a stronger form of the prime number theorem.

Penrose's theorem is more restricted, it only holds when matter obeys a stronger energy condition, called the dominant energy condition, which means that the energy is bigger than the pressure.

If f is a non-constant entire function, then its image is dense in C. This might seem to be a much stronger result than Liouville's theorem, but it is actually an easy corollary.

For each theorem that can be stated in the base system but is not provable in the base system, the goal is to determine the particular axiom system ( stronger than the base system ) that is necessary to prove that theorem.

In general, the Löwenheim – Skolem theorem does not hold in stronger logics such as second-order logic.

The modern statement of the theorem is both more general and stronger than the version for countable signatures stated in the introduction.

The Coase theorem considers all four of these outcomes logical because the economic incentives will be stronger than legal incentives.