** Daniel Liebowitz ; Charles Pearson: The Last Expedition: Stanley's Mad Journey Through the Congo, 2005, ISBN 0-393-05903-0

** Community ( reciprocity )

** Quadratic reciprocity, a fundamental result in number theory

** Cubic reciprocity, theorems that state conditions under which the congruence x < sup > 3 </ sup > ≡ p ( mod q ) is solvable

** Quartic reciprocity, a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x < sup > 4 </ sup > ≡ p ( mod q ) is solvable

** Artin reciprocity law, a general theorem in number theory that provided a partial solution to Hilbert's ninth problem

** April 8-He becomes the first to prove the quadratic reciprocity law, enabling determination of the solvability of any quadratic equation in modular arithmetic.

** Indirect reciprocity ( for example, reputation )

** Strong reciprocity

** Axiom 5: High levels of uncertainty produce high rates of reciprocity.

** Well-ordering theorem: Every set can be well-ordered.

** 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.

** If S is a set of sentences of first-order logic and B is a consistent subset of S, then B is included in a set that is maximal among consistent subsets of S. The special case where S is the set of all first-order sentences in a given signature is weaker, equivalent to the Boolean prime ideal theorem ; see the section " Weaker forms " below.

** The Vitali theorem on the existence of non-measurable sets which states that there is a subset of the real numbers that is not Lebesgue measurable.

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

** The Nielsen – Schreier theorem, that every subgroup of a free group is free.

** The Hahn – Banach theorem in functional analysis, allowing the extension of linear functionals

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

** The Banach – Alaoglu theorem about compactness of sets of functionals.

** The Baire category theorem about complete metric spaces, and its consequences, such as the open mapping theorem and the closed graph theorem.

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

** The numbers and are not algebraic numbers ( see the Lindemann – Weierstrass theorem ); hence they are transcendental.

** Hilbert's basis theorem

** Bayes ' theorem

** More generally, Rademacher's theorem extends the differentiability result to Lipschitz mappings between Euclidean spaces: a Lipschitz map ƒ: U → R < sup > m </ sup >, where U is an open set in R < sup > n </ sup >, is almost everywhere differentiable.

** Lyapunov's central limit theorem

** Superposition theorem, in electronics

** " Kelvin's vorticity theorem for incompressible or barotropic flow ".

** Various proofs of the four colour theorem.

* Stanley's reciprocity theorem

* Stanley's reciprocity theorem, states that a certain functional equation is satisfied by the generating function of any rational cone and the generating function of the cone's interior

In combinatorial mathematics, Stanley's reciprocity theorem, named after MIT mathematician Richard P. Stanley, states that a certain functional equation is satisfied by the generating function of any rational cone ( defined below ) and the generating function of the cone's interior.

Stanley's reciprocity theorem states that

* On Stanley's reciprocity theorem for rational cones, by Beck, Develin, and Robins -- arXiv abstract

The first generalization of the theorem is found in Gauss's second monograph ( 1832 ) on biquadratic reciprocity.

In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic which gives conditions for the solvability of quadratic equations modulo prime numbers.

There are a number of equivalent statements of the theorem, which consists of two " supplements " and the reciprocity law:

The Pythagorean theorem and the law of quadratic reciprocity are contenders for the title of theorem with the greatest number of distinct proofs.

This is a consequence of the reciprocity theorem of electromagnetics and is proved below.

Now for a given disposition of the antennas, the reciprocity theorem requires that the power transfer is equally effective in each direction, i. e.

The theorem of quadratic reciprocity ( which he had first succeeded in proving in 1796 ) relates the solvability of the congruence x < sup > 2 </ sup > ≡ q ( mod p ) to that of x < sup > 2 </ sup > ≡ p ( mod q ).

This is a consequence of the reciprocity theorem of electromagnetics.

Using Dirichlet's theorem on primes in arithmetic progressions, the law of quadratic reciprocity, and the Chinese remainder theorem ( CRT ) it is easy to see that for any M > 0 there are primes p such that the numbers 1, 2, …, M are all residues modulo p.

* Quadratic reciprocity, a theorem from number theory

He published important contributions to Fermat's last theorem, for which he proved the cases n = 5 and n = 14, and to the biquadratic reciprocity law.

In this case the reciprocity isomorphism of class field theory ( or Artin reciprocity map ) also admits an explicit description due to the Kronecker – Weber theorem.

The reciprocity theorem tells us that

As before, we can take advantage of the reciprocity theorem to provide a check on the accuracy of these measurements.

* Cubic reciprocity ( mathematics-number theory ), a theorem analogous to quadratic reciprocity

* Quadratic reciprocity, a theorem from modular arithmetic

By the Fundamental theorem of projective geometry a reciprocity is the composition of an automorphic function of K and a homography.