Local fieldIn mathematics, a field K is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation v and if its residue field k is finite. Equivalently, a local field is a locally compact topological field with respect to a non-discrete topology. Sometimes, real numbers R, and the complex numbers C (with their standard topologies) are also defined to be local fields; this is the convention we will adopt below.
Lexicographic orderIn mathematics, the lexicographic or lexicographical order (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the dictionaries to sequences of ordered symbols or, more generally, of elements of a totally ordered set. There are several variants and generalizations of the lexicographical ordering. One variant applies to sequences of different lengths by comparing the lengths of the sequences before considering their elements.
Schnirelmann densityIn additive number theory, the Schnirelmann density of a sequence of numbers is a way to measure how "dense" the sequence is. It is named after Russian mathematician Lev Schnirelmann, who was the first to study it. The Schnirelmann density of a set of natural numbers A is defined as where A(n) denotes the number of elements of A not exceeding n and inf is infimum. The Schnirelmann density is well-defined even if the limit of A(n)/n as n → ∞ fails to exist (see upper and lower asymptotic density).
Rough setIn computer science, a rough set, first described by Polish computer scientist Zdzisław I. Pawlak, is a formal approximation of a crisp set (i.e., conventional set) in terms of a pair of sets which give the lower and the upper approximation of the original set. In the standard version of rough set theory (Pawlak 1991), the lower- and upper-approximation sets are crisp sets, but in other variations, the approximating sets may be fuzzy sets. The following section contains an overview of the basic framework of rough set theory, as originally proposed by Zdzisław I.
Finite setIn mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, is a finite set with five elements. The number of elements of a finite set is a natural number (possibly zero) and is called the cardinality (or the cardinal number) of the set. A set that is not a finite set is called an infinite set.
Natural numberIn mathematics, the natural numbers are the numbers 1, 2, 3, etc., possibly including 0 as well. Some definitions, including the standard ISO 80000-2, begin the natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3, ..., whereas others start with 1, corresponding to the positive integers 1, 2, 3, ... Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, while in other writings, that term is used instead for the integers (including negative integers).
Rational varietyIn mathematics, a rational variety is an algebraic variety, over a given field K, which is birationally equivalent to a projective space of some dimension over K. This means that its function field is isomorphic to the field of all rational functions for some set of indeterminates, where d is the dimension of the variety. Let V be an affine algebraic variety of dimension d defined by a prime ideal I = ⟨f1, ..., fk⟩ in . If V is rational, then there are n + 1 polynomials g0, ..., gn in such that In order words, we have a of the variety.
Adele ringIn mathematics, the adele ring of a global field (also adelic ring, ring of adeles or ring of adèles) is a central object of class field theory, a branch of algebraic number theory. It is the restricted product of all the completions of the global field and is an example of a self-dual topological ring. An adele derives from a particular kind of idele. "Idele" derives from the French "idèle" and was coined by the French mathematician Claude Chevalley. The word stands for 'ideal element' (abbreviated: id.el.
Equations defining abelian varietiesIn mathematics, the concept of abelian variety is the higher-dimensional generalization of the elliptic curve. The equations defining abelian varieties are a topic of study because every abelian variety is a projective variety. In dimension d ≥ 2, however, it is no longer as straightforward to discuss such equations. There is a large classical literature on this question, which in a reformulation is, for complex algebraic geometry, a question of describing relations between theta functions.
Universal setIn set theory, a universal set is a set which contains all objects, including itself. In set theory as usually formulated, it can be proven in multiple ways that a universal set does not exist. However, some non-standard variants of set theory include a universal set. Many set theories do not allow for the existence of a universal set. There are several different arguments for its non-existence, based on different choices of axioms for set theory. In Zermelo–Fraenkel set theory, the axiom of regularity and axiom of pairing prevent any set from containing itself.
