This is the Bolyai-Gerwien theorem.

See also Bolyai-Gerwien theorem.

The theorem which we prove is more general than what we have described, since it works with the primary decomposition of the minimal polynomial, whether or not the primes which enter are all of first degree.

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.

This theorem is similar to the theorem of Kakutani that there exists a circumscribing cube around any closed, bounded convex set in Af.

According to Cauchy's functional equation theorem, the logarithm is the only continuous transformation that transforms real multiplication to addition.

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.

However, that particular case is a theorem of Zermelo – Fraenkel set theory without the axiom of choice ( ZF ); it is easily proved by mathematical induction.

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.

In Martin-Löf type theory and higher-order Heyting arithmetic, the appropriate statement of the axiom of choice is ( depending on approach ) included as an axiom or provable as a theorem.

Assuming ZF is consistent, Kurt Gödel showed that the negation of the axiom of choice is not a theorem of ZF by constructing an inner model ( the constructible universe ) which satisfies ZFC and thus showing that ZFC is consistent.

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.

It is possible, however, that there is a shorter proof of a theorem from ZFC than from ZF.

** Tarski's theorem: For every infinite set A, there is a bijective map between the sets A and A × A.

** König's theorem: Colloquially, the sum of a sequence of cardinals is strictly less than the product of a sequence of larger cardinals.

** Tychonoff's theorem stating that every product of compact topological spaces is compact.

Antiderivatives are related to definite integrals through the fundamental theorem of calculus: the definite integral of a function over an interval is equal to the difference between the values of an antiderivative evaluated at the endpoints of the interval.

In mathematics terminology, the vector space of bras is the dual space to the vector space of kets, and corresponding bras and kets are related by the Riesz representation theorem.

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

Euler's conjecture is a disproved conjecture in mathematics related to Fermat's last theorem which was proposed by Leonhard Euler in 1769.

For a closely related theorem see the Bing metrization theorem.

To apply the central limit theorem ( and related statistics ) to a power law distributed variable it is first necessary to ensure that the distribution has an exponent that exceeds 2.

Fault trees are a logical inverse of success trees, and may be obtained by applying de Morgan's theorem to success trees ( which are directly related to reliability block diagrams ).

The significance of the virial theorem is that it allows the average total kinetic energy to be calculated even for very complicated systems that defy an exact solution, such as those considered in statistical mechanics ; this average total kinetic energy is related to the temperature of the system by the equipartition theorem.

In the former case we have, which is related to the central limit theorem.

Zero – sum games are most often solved with the minimax theorem which is closely related to linear programming duality, or with Nash equilibrium.

The formula is further related to the particle's decay rate by the optical theorem.

Several theorems related to the triangle were known, including the binomial theorem.

Bayes ' rule is derived from and closely related to Bayes ' theorem.

The implicit function theorem is closely related to the inverse function theorem, which states when a function looks like graphs of invertible functions pasted together.

His other contributions include a simplified proof of the positive energy theorem involving spinors in general relativity, his work relating supersymmetry and Morse theory, his introduction of topological quantum field theory and related work on mirror symmetry, knot theory, twistor theory and D-branes and their intersections.

This task, known as automatic proof verification, is closely related to automated theorem proving.

The integral of the Gaussian curvature over the whole surface is closely related to the surface's Euler characteristic ; see the Gauss – Bonnet theorem.

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

The law can be expressed mathematically using vector calculus in integral form and differential form, both are equivalent since they are related by the divergence theorem, also called Gauss's theorem.

They also include statements less obviously related to compactness, such as the De Bruijn – Erdős theorem stating that every minimal k-chromatic graph is finite, and the Curtis – Hedlund – Lyndon theorem providing a topological characterization of cellular automata.

* The case n = 1 can be illustrated by the claim that there always exist a pair of opposite points on the earth's equator with the same temperature, this can be shown to be true much more easily using the intermediate value theorem.

In graph theory, much research was motivated by attempts to prove the four color theorem, first stated in 1852, but not proved until 1976 ( by Kenneth Appel and Wolfgang Haken, using substantial computer assistance ).

Experimentally, however, the total kinetic energy is found to be much greater: in particular, assuming the gravitational mass is due to only the visible matter of the galaxy, stars far from the center of galaxies have much higher velocities than predicted by the virial theorem.

It improves Dirichlet's theorem on prime numbers in arithmetic progressions, by showing that by averaging over the modulus over a range, the mean error is much less than can be proved in a given case.

Although the logical consequence relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic.

A more satisfying converse, which is much harder to prove, is the Looman – Menchoff theorem: if ƒ is continuous, u and v have first partial derivatives ( but not necessarily continuous ), and they satisfy the Cauchy – Riemann equations, then ƒ is holomorphic.

In particular, the fundamental theorem of calculus allows one to solve a much broader class of problems.

In 1930, Gödel's completeness theorem showed that propositional logic itself was complete in a much weaker sense — that is, any sentence that is unprovable from a given set of axioms must actually be false in some model of the axioms.

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

Because of Savitch's theorem, NPSPACE is equivalent to PSPACE, essentially because a deterministic Turing machine can simulate a nondeterministic Turing machine without needing much more space ( even though it may use much more time ).

In his work on the divine proportion, H. E. Huntley relates the feeling of reading and understanding someone else's proof of a theorem of mathematics to that of a viewer of a masterpiece of art — the reader of a proof has a similar sense of exhilaration at understanding as the original author of the proof, much as, he argues, the viewer of a masterpiece has a sense of exhilaration similar to the original painter or sculptor.

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.

A local embedding theorem is much simpler and can be proved using the implicit function theorem of advanced calculus.

The statement that ƒ can be reconstructed from ƒ ̂ is known as the Fourier integral theorem, and was first introduced in Fourier's Analytical Theory of Heat,, although what would be considered a proof by modern standards was not given until much later.

In Hilbert's axiomatization of geometry this statement is given as a theorem, but only after much groundwork.

The central limit theorem holds that, if there is sufficiently much scatter, the channel impulse response will be well-modelled as a Gaussian process irrespective of the distribution of the individual components.

For any collection of fixed size, the expected running time of the algorithm is finite for much the same reason that the infinite monkey theorem holds: there is some probability of getting the right permutation, so given an unbounded number of tries it will almost surely eventually be chosen.

It should not be confused with the space of ( bounded ) invertible operators on a Hilbert space, which is a larger group, and topologically much simpler, namely contractible — see Kuiper's theorem.

They arise in a much larger class of games because of the Sprague – Grundy theorem.