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| x Pi |
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Pi or π is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in Euclidean space; this is the same value as the ratio of a circle's area to the square of its radius. The symbol π was first proposed by the...
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| x Algebraically closed field |
In mathematics, a field F is said to be algebraically closed if every polynomial in one variable of degree at least 1, with coefficients in F, has a root in F.
As an example, the field of real numbers is not algebraically closed, because the...
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| x Algebraic number |
In mathematics, an algebraic number is a complex number that is a root of a non-zero polynomial in one variable with rational (or equivalently, integer) coefficients. Numbers such as π that are not algebraic are said to be transcendental, and are...
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| x Automorphism |
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of...
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| x Antisymmetric relation |
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In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in X
or, equivalently,
In mathematical notation, this is:
or equally,
An example of an antisymmetric relation is the subset relation:
Or in words, if every element...
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| x Associativity |
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In mathematics, associativity is a property that a binary operation can have. It means that, within an expression containing two or more occurrences in a row of the same associative operator, the order that the operations are performed does not...
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| x Associative algebra |
In mathematics, an associative algebra is a module which also allows the multiplication of vectors in a distributive and associative manner. They are thus a special case of algebras over commutative rings.
Let R be a fixed commutative ring. An...
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| x Abelian group |
An abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order (the axiom of commutativity). Abelian groups generalize the arithmetic of addition...
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| x Arithmetic |
Arithmetic or arithmetics (from the Greek word αριθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations, such...
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| x Algebraic closure |
In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics.
Using Zorn's lemma, it can be shown that every field has an...
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| x Arithmetic function |
In number theory, an arithmetic (or arithmetical) function is a real or complex valued function ƒ(n) defined on the set of natural numbers (i.e. positive integers) that "expresses some arithmetical property of n."
An example of an arithmetic...
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| x Banach space |
In mathematics, Banach spaces (pronounced [ˈbanax]) are one of the central objects of study in functional analysis. Many of the infinite-dimensional function spaces studied in analysis are Banach spaces, including spaces of continuous functions ...
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| x Bilinear operator |
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In mathematics, a bilinear operator is a function combining elements of two vector space to yield an element of third vector space that is linear in each of its arguments. Matrix multiplication is an example.
Let V, W and X be three vector spaces...
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| x Banach algebra |
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space. The algebra multiplication and the Banach space...
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| x Cauchy sequence |
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In mathematics, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. To be more precise, by dropping enough (but still only a finite number of) terms from the...
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| x Connected space |
In topology and related branches of mathematics, a connected space is a topological space which cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is...
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| x Cofinality |
In mathematics, especially in order theory, the cofinality cf(A) of a partially ordered set A is the least of the cardinalities of the cofinal subsets of A.
This definition of cofinality relies on the axiom of choice, as it uses the fact that every...
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| x C*-algebra |
C*-algebras (pronounced "C-star") are an important area of research in functional analysis, a branch of mathematics. The prototypical example of a C*-algebra is a complex algebra A of linear operators on a complex Hilbert space with two additional...
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| x Dual space |
In mathematics, any vector space, V, has a corresponding dual vector space (or just dual space for short) consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors...
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| x Ordinary differential equation |
In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable.
A simple example is Newton's second law of motion,...
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| x Diffeomorphism |
In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.
Given two manifolds M and...
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| x Commutator subgroup |
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.
The commutator subgroup is important because it is the smallest normal...
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| x Direct product |
In mathematics, one can often define a direct product of objects already known, giving a new one. This is generally the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one...
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| x Directed set |
In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation ≤ (that is, a preorder), with the additional property that every pair of elements has an upper...
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| x Equivalence class |
In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a:
The notion of equivalence classes is useful for constructing sets out of...
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| x Elementary group theory |
In mathematics, a group is defined as a non-empty set G and a binary operation called the group operation.
As a shortcut is noted or even xy. This is called infix notation.
The group must obey the following rules (or axioms). Let a,b,c be...
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| x Elementary algebra |
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Elementary algebra is a fundamental and relatively basic form of algebra taught to students who are presumed to have little or no formal knowledge of mathematics beyond arithmetic. While in arithmetic only numbers and their arithmetical operations ...
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| x Functor |
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms in the category of small categories.
Functors were first considered...
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| x Fundamental group |
In mathematics, more specifically algebraic topology, the fundamental group or Poincaré group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point,...
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| x Quotient group |
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In mathematics, specifically group theory, a quotient group (or factor group) is a group obtained by identifying together elements of a larger group using an equivalence relation. For example, the cyclic group of addition modulo n can be obtained...
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| x Field extension |
In mathematics, more specifically in abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and...
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| x Groupoid |
In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group and of category in several equivalent ways. A groupoid can be seen as a:
Special cases...
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| x Galois group |
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In mathematics, a Galois group is a group associated with a certain type of field extension. The study of field extensions (and polynomials which give rise to them) via Galois groups is called Galois theory after Évariste Galois who first invented...
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| x Group representation |
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group...
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| x Group action |
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set and the symmetries of the object are described by the symmetry group of this set, which...
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| x Grothendieck topology |
In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a...
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| x Hausdorff dimension |
In mathematics, the Hausdorff dimension (also known as the Hausdorff–Besicovitch dimension) is an extended non-negative real number associated to any metric space. The Hausdorff dimension generalizes the notion of the dimension of a real vector...
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| x Harmonic mean |
In mathematics, the harmonic mean (formerly sometimes called the subcontrary mean) is one of several kinds of average. Typically, it is appropriate for situations when the average of rates is desired.
The harmonic mean H of the positive real numbers...
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| x Inner product space |
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In mathematics, an inner product space is a vector space with the additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors....
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| x If and only if |
In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements. In that it is biconditional, the connective can be likened to the standard material conditional ...
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| x Identity element |
In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them. This is used for groups and related concepts....
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| x Identity function |
In mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that was used as its argument. In terms of equations, the function is given by f(x) = x.
Formally, if M is a...
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| x Inverse limit |
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In mathematics, the inverse limit (also called the projective limit) is a construction which allows one to "glue together" several related objects, the precise manner of the gluing process being specified by morphisms between the objects. Inverse...
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| x Series |
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In mathematics, given an infinite sequence of numbers { an }, a series is informally the result of adding all those terms together: a1 + a2 + a3 + · · ·. These can be written more compactly using the summation symbol ∑. An example is the famous...
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| x Integral domain |
In abstract algebra, an integral domain is a commutative ring without zero divisors and with a multiplicative identity 1 not equal to 0, the additive identity. Integral domains are generalizations of the integers and provide a natural setting for...
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| x Lie algebra |
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In mathematics, a Lie algebra (pronounced /ˈliː/ ("lee"), not /ˈlaɪ/ ("lye")) is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the...
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| x Linear transformation |
In mathematics, a linear map (also called a linear transformation, linear function or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. The expression "linear...
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| x Lebesgue measure |
In mathematics, the Lebesgue measure, named after Henri Lebesgue, is the standard way of assigning a length, area or volume to subsets of Euclidean space. It is used throughout real analysis, in particular to define Lebesgue integration. Sets which...
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| x Morphism |
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures.
The study of morphisms and of the structures (called objects) over which they are defined, is central to category theory....
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| x Monoid |
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids...
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| x Filter |
In mathematics, a filter is a special subset of a partially ordered set. A frequently used special case is the situation that the ordered set under consideration is just the power set of some set, ordered by set inclusion. Filters appear in order...
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| x Measure |
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In mathematics, more specifically in measure theory, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts...
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| x Normal subgroup |
In mathematics, more specifically in abstract algebra, a normal subgroup is a special kind of subgroup. Normal subgroups are important because they can be used to construct quotient groups from a given group.
Évariste Galois was the first to realize...
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| x Net |
In mathematics, more specifically in point-set topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence. In essence, a sequence is a function with domain the natural numbers, and in the context of...
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| x Partially ordered set |
In mathematics, especially order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that...
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| x Pro-finite group |
In mathematics, profinite groups are topological groups that are in a certain sense assembled from finite groups; they share many properties with their finite quotients.
Formally, a profinite group is a Hausdorff, compact, and totally disconnected...
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| x Projective plane |
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In mathematics, a projective plane has two possible definitions, one of them coming from linear algebra, and another (which is more general) coming from axiomatic and finite geometry. The first definition quickly produces planes that are homogeneous...
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| x Permutation group |
In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose group operation is the composition of permutations in G (which are thought of as bijective functions from the set M to itself); the...
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| x P-group |
In mathematics, given a prime number p, a p-group is a periodic group in which each element has a power of p as its order. That is, for each element g of the group, there exists a nonnegative integer n such that g to the power p is equal to the...
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| x Product topology |
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In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more obvious, topology...
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