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The statement that this is the only quadratic pairing function is known as the Fueter–Pólya theorem. [9] Whether this is the only polynomial pairing function is still an open question. When we apply the pairing function to k 1 and k 2 we often denote the resulting number as k 1, k 2 . [citation needed]
In mathematics, a relation denotes some kind of relationship between two objects in a set, which may or may not hold. [1] As an example, " is less than " is a relation on the set of natural numbers ; it holds, for instance, between the values 1 and 3 (denoted as 1 < 3 ), and likewise between 3 and 4 (denoted as 3 < 4 ), but not between the ...
A Langford pairing for n = 4. In combinatorial mathematics, a Langford pairing, also called a Langford sequence, is a permutation of the sequence of 2n numbers 1, 1, 2, 2, ..., n, n in which the two 1s are one unit apart, the two 2s are two units apart, and more generally the two copies of each number k are k units apart. Langford pairings are ...
In arithmetic geometry, the Tate–Shafarevich group ะจ(A/K) of an abelian variety A (or more generally a group scheme) defined over a number field K consists of the elements of the Weil–Châtelet group (/) = (,), where = (/) is the absolute Galois group of K, that become trivial in all of the completions of K (i.e., the real and complex completions as well as the p-adic fields obtained from ...
The Weil pairing is an important concept in elliptic curve cryptography; e.g., it may be used to attack certain elliptic curves (see MOV attack). It and other pairings have been used to develop identity-based encryption schemes.
Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory.Geometric, algebraic, and arithmetic objects are assigned objects called K-groups.
Each scheme X over k determines two objects in DM called the motive of X, M(X), and the compactly supported motive of X, M c (X); the two are isomorphic if X is proper over k. One basic point of the derived category of motives is that the four types of motivic homology and motivic cohomology all arise as sets of morphisms in this category.
In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x = 0 and x 2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers).