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Mathematical Concept table
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| x name | x image | x Also Typed With | x Discoverer | x article |
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| x Algebraically closed field | Concepts/Theories |
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 | Concepts/Theories |
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. Complex numbers such as pi that are not algebraic are said to be transcendental,...
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| x Automorphism | Concepts/Theories |
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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Concepts/Theories |
In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in X, if a is R to b and b is R to a, then a = b.
In mathematical notation, this is:
or equally,
Inequalities are antisymmetric, since for numbers a and b, a ≤ b and...
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| x Associativity |
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Concepts/Theories |
In mathematics, associativity is a property that a binary operation can have. It means that, within an expression containing two or more of the same associative operators in a row, the order that the operations occur does not matter as long as the...
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| x Associative algebra | Concepts/Theories |
In mathematics, an associative algebra is a vector space (or more generally, a module) which also allows the multiplication of vectors in a distributive and associative manner. They are thus special algebras.
An associative algebra A over a field K...
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| x Abelian group | Concepts/Theories |
An abelian group, also called a commutative group, is a group satisfying the requirement that the product of elements does not depend on their order (the axiom of commutativity). Abelian groups generalize the arithmetic of addition of integers; they...
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| x Arithmetic | Concepts/Theories |
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. In...
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| x Algebraic closure | Concepts/Theories |
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 | Concepts/Theories |
In number theory an arithmetic function or arithmetical function is a function defined on the set of natural numbers (i.e. positive integers) that takes real or complex values. A simple example of an arithmetic function is the characteristic...
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| x Banach space | Concepts/Theories |
In mathematics, Banach spaces (pronounced [ˈbanax], named after Polish mathematician Stefan Banach) are one of the central objects of study in functional analysis. They are topological vector spaces that have many interesting properties associated...
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| x Bilinear operator | Concepts/Theories |
In mathematics, a bilinear map is a function of two arguments that is linear in each. An example of such a map is multiplication of integers.
Let V, W and X be three vector spaces over the same base field F. A bilinear map is a function
such that...
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| x Banach algebra | Concepts/Theories |
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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Concepts/Theories |
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 | Concepts/Theories |
In topology and related branches of mathematics, a connected space is a topological space which cannot be represented as the disjoint union of two or more nonempty open subsets. Connectedness is one of the principal topological properties that is...
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| x Cofinality | Concepts/Theories |
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 | Concepts/Theories |
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 | Concepts/Theories |
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 which...
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| x Ordinary differential equation | Concepts/Theories |
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 | Concepts/Theories |
In mathematics, a diffeomorphism is an isomorphism 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 N, a bijective...
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| x Commutator subgroup | Concepts/Theories |
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 | Concepts/Theories |
In mathematics, one can often define a direct product of objects already known, giving a new one. Examples are the product of sets (see Cartesian product), groups (described below), the product of rings and of other algebraic structures. The product...
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| x Directed set | Concepts/Theories |
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 (i.e., preorder) ≤ having the additional property that every pair of elements has an upper bound;...
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| x Equivalence class | Concepts/Theories |
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 | Concepts/Theories |
In mathematics, a group is defined as a set G and a binary operation * on G, called product and denoted by infix "*". The operation obeys the following rules (also called axioms). Let a, b, and c be arbitrary elements of G. Then:
An abelian group...
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| x Elementary algebra | Concepts/Theories |
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 | Concepts/Theories |
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as morphisms in the category of small categories.
Functors were first considered in algebraic topology, where...
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| x Fundamental group | Concepts/Theories |
In mathematics, the fundamental group is one of the basic concepts of algebraic topology. Associated with every point of a topological space there is a fundamental group that conveys information about the 1-dimensional structure of the portion of...
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| x Quotient group |
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Concepts/Theories |
In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is intuitively a group that "collapses" the normal subgroup N to the identity element. The quotient group is written G/N and is usually...
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| x Field extension | Concepts/Theories |
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 | Concepts/Theories |
In mathematics, especially in category theory and homotopy theory, a groupoid is a simultaneous generalisation of a group, a setoid (a set equipped with an equivalence relation), and a G-set (a set equipped with an action of a group G). Groupoids...
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| x Galois group |
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Concepts/Theories |
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 | Concepts/Theories |
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 | Concepts/Theories |
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 | Concepts/Theories |
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 | Concepts/Theories |
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 | Concepts/Theories |
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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Concepts/Theories |
In mathematics, an inner product space is a vector space with the additional structure of 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. Inner...
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| x If and only if | Concepts/Theories |
If and only if, in logic and fields that rely on it such as mathematics and philosophy, is a biconditional logical connective between statements which means that the truth of either one of the statements requires the truth of the other. Thus, either...
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| x Identity element | Concepts/Theories |
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 | Concepts/Theories |
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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Concepts/Theories |
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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Concepts/Theories |
In mathematics, a series is often represented as the sum of a sequence of terms. That is, a series is represented as a list of numbers with addition operations between them, for example this arithmetic sequence:
In most cases of interest the terms...
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| x Integral domain | Concepts/Theories |
In abstract algebra, a branch of mathematics, an integral domain is a commutative ring with an additive identity 0 and a multiplicative identity 1 such that 0 ≠ 1, in which the product of any two non-zero elements is always non-zero (the zero...
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| x Lie algebra |
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Concepts/Theories |
In mathematics, a Lie algebra 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 concept of infinitesimal transformations. The term ...
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