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Left-unique relation Summary Right-total_relation In mathematics, a binary relation associates elements of one set, called the domain, with elements of another set, called the codomain. A binary relation over sets X and Y is a new set of ordered pairs (x, y) consisting of elements x in X and y in Y. It is a generalizat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Left-unique relation Summary Right-total_relation {\displaystyle X_{1}\times \cdots \times X_{n}.} An example of a binary relation is the "divides" relation over the set of prime numbers P {\displaystyle \mathbb {P} } and the set of integers Z {\displaystyle \mathbb {Z} } , in which each prime p is related to each inte... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Left-unique relation Summary Right-total_relation These include, among others: the "is greater than", "is equal to", and "divides" relations in arithmetic; the "is congruent to" relation in geometry; the "is adjacent to" relation in graph theory; the "is orthogonal to" relation in linear algebra.A function may be defin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Left-unique relation Summary Right-total_relation A binary relation over sets X and Y is an element of the power set of X × Y . {\displaystyle X\times Y.} Since the latter set is ordered by inclusion (⊆), each relation has a place in the lattice of subsets of X × Y . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Left-unique relation Summary Right-total_relation {\displaystyle X\times Y.} A binary relation is called a homogeneous relation when X = Y. A binary relation is also called a heterogeneous relation when it is not necessary that X = Y. Since relations are sets, they can be manipulated using set operations, including uni... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Left-unique relation Summary Right-total_relation Beyond that, operations like the converse of a relation and the composition of relations are available, satisfying the laws of a calculus of relations, for which there are textbooks by Ernst Schröder, Clarence Lewis, and Gunther Schmidt. A deeper analysis of relations i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Relation (mathematics) Summary Relation_(mathematics) In mathematics, a binary relation on a set may, or may not, hold between two given set members. For example, "is less than" is a relation on the set of natural numbers; it holds e.g. between 1 and 3 (denoted as 1<3) , and likewise between 3 and 4 (denoted as 3<4), b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Relation (mathematics) Summary Relation_(mathematics) Formally, a relation R over a set X can be seen as a set of ordered pairs (x, y) of members of X. The relation R holds between x and y if (x, y) is a member of R. For example, the relation "is less than" on the natural numbers is an infinite set Rless of pairs of na... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Relation (mathematics) Summary Relation_(mathematics) For most common relations in mathematics, special symbols are introduced, like "<" for "is less than", and "|" for "is a nontrivial divisor of", and, most popular "=" for "is equal to". For example, "1<3", "1 is less than 3", and "(1,3) ∈ Rless" mean all the same; s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Relation (mathematics) Summary Relation_(mathematics) Various properties of relations are investigated. A relation R is reflexive if xRx holds for all x, and irreflexive if xRx holds for no x. It is symmetric if xRy always implies yRx, and asymmetric if xRy implies that yRx is impossible. It is transitive if xRy and yR... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Relation (mathematics) Summary Relation_(mathematics) For example, "is less than" is irreflexive, asymmetric, and transitive, but neither reflexive nor symmetric. "is sister of" is transitive, but neither reflexive (e.g. Pierre Curie is not a sister of himself), nor symmetric, nor asymmetric; while being irreflexive or... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Relation (mathematics) Summary Relation_(mathematics) Mathematical theorems are known about combinations of relation properties, such as "A transitive relation is irreflexive if, and only if, it is asymmetric". Of particular importance are relations that satisfy certain combinations of properties. A partial order is a ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Binomial ring Summary Binomial_ring In mathematics, a binomial ring is a commutative ring whose additive group is torsion-free and contains all binomial coefficients ( x n ) = x ( x − 1 ) ⋯ ( x − n + 1 ) n ! {\displaystyle {\binom {x}{n}}={\frac {x(x-1)\cdots (x-n+1)}{n!}}} for x in the ring and n a positive integer. B... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Biorthogonal polynomials Summary Biorthogonal_polynomial In mathematics, a biorthogonal polynomial is a polynomial that is orthogonal to several different measures. Biorthogonal polynomials are a generalization of orthogonal polynomials and share many of their properties. There are two different concepts of biorthogona... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Biorthogonal system Summary Biorthogonal_system In mathematics, a biorthogonal system is a pair of indexed families of vectors such that where E {\displaystyle E} and F {\displaystyle F} form a pair of topological vector spaces that are in duality, ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \,\cdot ,\cdot \,\rangle } is a biline... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bipartite matroid Summary Bipartite_matroid In mathematics, a bipartite matroid is a matroid all of whose circuits have even size. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Biquadratic field Summary Biquadratic_field In mathematics, a biquadratic field is a number field K of a particular kind, which is a Galois extension of the rational number field Q with Galois group the Klein four-group. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Biquaternion algebra Summary Biquaternion_algebra In mathematics, a biquaternion algebra is a compound of quaternion algebras over a field. The biquaternions of William Rowan Hamilton (1844) and the related split-biquaternions and dual quaternions do not form biquaternion algebras in this sense. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bishop's graph Summary Bishop's_graph In mathematics, a bishop's graph is a graph that represents all legal moves of the chess piece the bishop on a chessboard. Each vertex represents a square on the chessboard and each edge represents a legal move of the bishop; that is, there is an edge between two vertices (squares)... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bisymmetric matrix Summary Bisymmetric_matrix In mathematics, a bisymmetric matrix is a square matrix that is symmetric about both of its main diagonals. More precisely, an n × n matrix A is bisymmetric if it satisfies both A = AT and AJ = JA where J is the n × n exchange matrix. For example, any matrix of the form = ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bitopological space Summary Bitopological_space In mathematics, a bitopological space is a set endowed with two topologies. Typically, if the set is X {\displaystyle X} and the topologies are σ {\displaystyle \sigma } and τ {\displaystyle \tau } then the bitopological space is referred to as ( X , σ , τ ) {\displaystyl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bivariant Chow group Summary Bivariant_theory In mathematics, a bivariant theory was introduced by Fulton and MacPherson (Fulton & MacPherson 1981), in order to put a ring structure on the Chow group of a singular variety, the resulting ring called an operational Chow ring. On technical levels, a bivariant theory is a ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bivector (complex) Summary Bivector_(complex) In mathematics, a bivector is the vector part of a biquaternion. For biquaternion q = w + xi + yj + zk, w is called the biscalar and xi + yj + zk is its bivector part. The coordinates w, x, y, z are complex numbers with imaginary unit h: x = x 1 + h x 2 , y = y 1 + h y 2 , ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bivector (complex) Summary Bivector_(complex) A bivector may be written as the sum of real and imaginary parts: ( x 1 i + y 1 j + z 1 k ) + h ( x 2 i + y 2 j + z 2 k ) {\displaystyle (x_{1}\mathrm {i} +y_{1}\mathrm {j} +z_{1}\mathrm {k} )+\mathrm {h} (x_{2}\mathrm {i} +y_{2}\mathrm {j} +z_{2}\mathrm {k} )} where r 1 = ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bivector (complex) Summary Bivector_(complex) The Lie algebra of the Lorentz group is expressed by bivectors. In particular, if r1 and r2 are right versors so that r 1 2 = − 1 = r 2 2 {\displaystyle r_{1}^{2}=-1=r_{2}^{2}} , then the biquaternion curve {exp θr1: θ ∈ R} traces over and over the unit circle in the plane ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bivector (complex) Summary Bivector_(complex) Now (hr2)2 = (−1)(−1) = +1, and the biquaternion curve {exp θ(hr2): θ ∈ R} is a unit hyperbola in the plane {x + yr2: x, y ∈ R}. The spacetime transformations in the Lorentz group that lead to FitzGerald contractions and time dilation depend on a hyperbolic angle parameter.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bivector (complex) Summary Bivector_(complex) "The commutator product of this Lie algebra is just twice the cross product on R3, for instance, = ij − ji = 2k, which is twice i × j. As Shaw wrote in 1970: Now it is well known that the Lie algebra of the homogeneous Lorentz group can be considered to be that of bivector... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bivector (complex) Summary Bivector_(complex) : 665 The popular text Vector Analysis (1901) used the term. : 249 Given a bivector r = r1 + hr2, the ellipse for which r1 and r2 are a pair of conjugate semi-diameters is called the directional ellipse of the bivector r.: 436 In the standard linear representation of biquat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bivector (complex) Summary Bivector_(complex) Ludwik Silberstein studied a complexified electromagnetic field E + hB, where there are three components, each a complex number, known as the Riemann–Silberstein vector. "Bivectors help describe elliptically polarized homogeneous and inhomogeneous plane waves – one vector ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bivector Summary Bivector In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. If a scalar is considered a degree-zero quantity, and a vector is a degree-one quantity, then a bivector can be thought of as being of degree two. Bivecto... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bivector Summary Bivector They can be used to generate rotations in any number of dimensions, and are a useful tool for classifying such rotations. They are also used in physics, tying together a number of otherwise unrelated quantities. Geometrically, a simple bivector can be interpreted as an oriented plane segment, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bivector Summary Bivector The bivector a ∧ b has a magnitude equal to the area of the parallelogram with edges a and b, has the orientation (or attitude) of the plane spanned by a and b, and has orientation being the sense of the rotation that would align a with b. In layman terms, any surface is the same bivector, if ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Partitioned matrix Summary Partitioned_matrix In mathematics, a block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original matrix with a collection of horizon... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Partitioned matrix Summary Partitioned_matrix This notion can be made more precise for an n {\displaystyle n} by m {\displaystyle m} matrix M {\displaystyle M} by partitioning n {\displaystyle n} into a collection rowgroups {\displaystyle {\text{rowgroups}}} , and then partitioning m {\displaystyle m} into a collection... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Block matrix pseudoinverse Summary Block_matrix_pseudoinverse In mathematics, a block matrix pseudoinverse is a formula for the pseudoinverse of a partitioned matrix. This is useful for decomposing or approximating many algorithms updating parameters in signal processing, which are based on the least squares method. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bouquet graph Summary Bouquet_graph In mathematics, a bouquet graph B m {\displaystyle B_{m}} , for an integer parameter m {\displaystyle m} , is an undirected graph with one vertex and m {\displaystyle m} edges, all of which are self-loops. It is the graph-theoretic analogue of the topological bouquet, a space of m {\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bouquet graph Summary Bouquet_graph In particular, every cellularly embedded graph can be reduced to an embedded bouquet by a partial duality applied to the edges of any spanning tree of the graph, or alternatively by contracting the edges of any spanning tree. In graph-theoretic approaches to group theory, every Cayle... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Boxcar function Summary Boxcar_function In mathematics, a boxcar function is any function which is zero over the entire real line except for a single interval where it is equal to a constant, A. The function is named after its graph's resemblance to a boxcar, a type of railroad car. The boxcar function can be expressed... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bracket algebra Summary Bracket_algebra In mathematics, a bracket algebra is an algebraic system that connects the notion of a supersymmetry algebra with a symbolic representation of projective invariants. Given that L is a proper signed alphabet and Super is the supersymmetric algebra, the bracket algebra Bracket of d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Braided Hopf algebra Summary Braided_Hopf_algebra In mathematics, a braided Hopf algebra is a Hopf algebra in a braided monoidal category. The most common braided Hopf algebras are objects in a Yetter–Drinfeld category of a Hopf algebra H, particularly the Nichols algebra of a braided vector space in that category. The... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Braided vector space Summary Braided_vector_space In mathematics, a braided vector space V {\displaystyle \;V} is a vector space together with an additional structure map τ {\displaystyle \tau } symbolizing interchanging of two vector tensor copies: τ: V ⊗ V ⟶ V ⊗ V {\displaystyle \tau :\;V\otimes V\longrightarrow V\ot... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Braided vector space Summary Braided_vector_space A superspace has a braiding with negative sign in braiding two odd vectors. More generally, a diagonal braiding means that for a V {\displaystyle \;V} -base x i {\displaystyle x_{i}} we have τ ( x i ⊗ x j ) = q i j ( x j ⊗ x i ) {\displaystyle \tau (x_{i}\otimes x_{j})=... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ramified cover Summary Ramified_cover In mathematics, a branched covering is a map that is almost a covering map, except on a small set. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Branched manifold Summary Branched_manifold In mathematics, a branched manifold is a generalization of a differentiable manifold which may have singularities of very restricted type and admits a well-defined tangent space at each point. A branched n-manifold is covered by n-dimensional "coordinate charts", each of whic... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Branched surface Summary Branched_surface In mathematics, a branched surface is a generalization of both surfaces and train tracks. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Brownian sheet Summary Brownian_sheet In mathematics, a brownian sheet is a multiparametric generalization of the brownian motion to a gaussian random field. This means we generalize the "time" parameter t {\displaystyle t} of a brownian motion B t {\displaystyle B_{t}} from R + {\displaystyle \mathbb {R} _{+}} to R + ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Building (mathematics) Summary Building_(mathematics) In mathematics, a building (also Tits building, named after Jacques Tits) is a combinatorial and geometric structure which simultaneously generalizes certain aspects of flag manifolds, finite projective planes, and Riemannian symmetric spaces. Buildings were initial... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bullet-nose curve Summary Bullet-nose_curve In mathematics, a bullet-nose curve is a unicursal quartic curve with three inflection points, given by the equation a 2 y 2 − b 2 x 2 = x 2 y 2 {\displaystyle a^{2}y^{2}-b^{2}x^{2}=x^{2}y^{2}\,} The bullet curve has three double points in the real projective plane, at x = 0 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Test function Summary Test_function In mathematics, a bump function (also called a test function) is a function f: R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } on a Euclidean space R n {\displaystyle \mathbb {R} ^{n}} which is both smooth (in the sense of having continuous derivatives of all orders) and c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bundle gerbe Summary Bundle_gerbe In mathematics, a bundle gerbe is a geometrical model of certain 1-gerbes with connection, or equivalently of a 2-class in Deligne cohomology. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bundle (mathematics) Summary Bundle_(mathematics) In mathematics, a bundle is a generalization of a fiber bundle dropping the condition of a local product structure. The requirement of a local product structure rests on the bundle having a topology. Without this requirement, more general objects can be considered bundl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bundle homomorphism Summary Bundle_homomorphism In mathematics, a bundle map (or bundle morphism) is a morphism in the category of fiber bundles. There are two distinct, but closely related, notions of bundle map, depending on whether the fiber bundles in question have a common base space. There are also several variat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cancellative semigroup Summary Cancellation_semigroup In mathematics, a cancellative semigroup (also called a cancellation semigroup) is a semigroup having the cancellation property. In intuitive terms, the cancellation property asserts that from an equality of the form a·b = a·c, where · is a binary operation, one can... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cancellative semigroup Summary Cancellation_semigroup Prototypical examples of cancellative semigroups are the positive integers under addition or multiplication. Cancellative semigroups are considered to be very close to being groups because cancellability is one of the necessary conditions for a semigroup to be embed... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Canonical basis Summary Canonical_basis In mathematics, a canonical basis is a basis of an algebraic structure that is canonical in a sense that depends on the precise context: In a coordinate space, and more generally in a free module, it refers to the standard basis defined by the Kronecker delta. In a polynomial rin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Canonical homomorphism Summary Canonical_map In mathematics, a canonical map, also called a natural map, is a map or morphism between objects that arises naturally from the definition or the construction of the objects. Often, it is a map which preserves the widest amount of structure. A choice of a canonical map somet... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Canonical homomorphism Summary Canonical_map These are also sometimes called canonical maps. A canonical isomorphism is a canonical map that is also an isomorphism (i.e., invertible). In some contexts, it might be necessary to address an issue of choices of canonical maps or canonical isomorphisms; for a typical exampl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Character (topology) Summary Cardinal_function In mathematics, a cardinal function (or cardinal invariant) is a function that returns cardinal numbers. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Superstrong cardinal Summary Superstrong_cardinal In mathematics, a cardinal number κ is called superstrong if and only if there exists an elementary embedding j: V → M from V into a transitive inner model M with critical point κ and V j ( κ ) {\displaystyle V_{j(\kappa )}} ⊆ M. Similarly, a cardinal κ is n-superstrong... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ω-huge cardinal Summary Huge_cardinal In mathematics, a cardinal number κ {\displaystyle \kappa } is called huge if there exists an elementary embedding j: V → M {\displaystyle j:V\to M} from V {\displaystyle V} into a transitive inner model M {\displaystyle M} with critical point κ {\displaystyle \kappa } and j ( κ ) ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cardinal addition Summary Cardinal_multiplication In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the case of infinite sets, the infinite... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cardinal addition Summary Cardinal_multiplication Two sets have the same cardinality if, and only if, there is a one-to-one correspondence (bijection) between the elements of the two sets. In the case of finite sets, this agrees with the intuitive notion of number of elements. In the case of infinite sets, the behavior... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cardinal addition Summary Cardinal_multiplication A fundamental theorem due to Georg Cantor shows that it is possible for infinite sets to have different cardinalities, and in particular the cardinality of the set of real numbers is greater than the cardinality of the set of natural numbers. It is also possible for a p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cardinal addition Summary Cardinal_multiplication {\displaystyle 0,1,2,3,\ldots ,n,\ldots ;\aleph _{0},\aleph _{1},\aleph _{2},\ldots ,\aleph _{\alpha },\ldots .\ } This sequence starts with the natural numbers including zero (finite cardinals), which are followed by the aleph numbers. The aleph numbers are indexed by ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cardinal addition Summary Cardinal_multiplication If the axiom of choice is not true (see Axiom of choice § Independence), there are infinite cardinals that are not aleph numbers. Cardinality is studied for its own sake as part of set theory. It is also a tool used in branches of mathematics including model theory, com... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Suslin cardinal Summary Suslin_cardinal In mathematics, a cardinal λ < Θ is a Suslin cardinal if there exists a set P ⊂ 2ω such that P is λ-Suslin but P is not λ'-Suslin for any λ' < λ. It is named after the Russian mathematician Mikhail Yakovlevich Suslin (1894–1919). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Categorical ring Summary 2-ring In mathematics, a categorical ring is, roughly, a category equipped with addition and multiplication. In other words, a categorical ring is obtained by replacing the underlying set of a ring by a category. For example, given a ring R, let C be a category whose objects are the elements of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Categorical ring Summary 2-ring Then C is a categorical ring. But the point is that one can also consider the situation in which an element of R comes with a "nontrivial automorphism" (cf. Lurie). This line of generalization of a ring eventually leads to the notion of an En-ring. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Category (mathematics) Summary Category_(mathematics) In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the exist... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Category (mathematics) Summary Category_(mathematics) Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. As such, category theory provides an alternative foundation for mathemat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Category (mathematics) Summary Category_(mathematics) In addition to formalizing mathematics, category theory is also used to formalize many other systems in computer science, such as the semantics of programming languages. Two categories are the same if they have the same collection of objects, the same collection of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Category (mathematics) Summary Category_(mathematics) Well-known categories are denoted by a short capitalized word or abbreviation in bold or italics: examples include Set, the category of sets and set functions; Ring, the category of rings and ring homomorphisms; and Top, the category of topological spaces and contin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Category (mathematics) Summary Category_(mathematics) Other references are given in the References below. The basic definitions in this article are contained within the first few chapters of any of these books. Any monoid can be understood as a special sort of category (with a single object whose self-morphisms are rep... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Distributive category Summary Distributive_category In mathematics, a category is distributive if it has finite products and finite coproducts and such that for every choice of objects A , B , C {\displaystyle A,B,C} , the canonical map : A × B + A × C → A × ( B + C ) {\displaystyle :A\!\times \!B\,+A\!\times \!C\to A\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Catholic semigroup Summary Catholic_semigroup In mathematics, a catholic semigroup is a semigroup in which no two distinct elements have the same set of inverses. The terminology was introduced by B. M. Schein in a paper published in 1979. Every catholic semigroup either is a regular semigroup or has precisely one elem... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Catholic semigroup Summary Catholic_semigroup The semigroup of all partial transformations of a set is a catholic semigroup. It follows that every semigroup is embeddable in a catholic semigroup. But the full transformation semigroup on a set is not catholic unless the set is a singleton set. Regular catholic semigroup... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cochain complexes Summary Category_of_chain_complexes In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of the next. Associated... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cochain complexes Summary Category_of_chain_complexes In algebraic topology, the singular chain complex of a topological space X is constructed using continuous maps from a simplex to X, and the homomorphisms of the chain complex capture how these maps restrict to the boundary of the simplex. The homology of this chain... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Change of variables Summary Change_of_variable In mathematics, a change of variables is a basic technique used to simplify problems in which the original variables are replaced with functions of other variables. The intent is that when expressed in new variables, the problem may become simpler, or equivalent to a bette... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Change of variables Summary Change_of_variable A very simple example of a useful variable change can be seen in the problem of finding the roots of the sixth-degree polynomial: x 6 − 9 x 3 + 8 = 0. {\displaystyle x^{6}-9x^{3}+8=0.} Sixth-degree polynomial equations are generally impossible to solve in terms of radicals... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Change of variables Summary Change_of_variable This particular equation, however, may be written ( x 3 ) 2 − 9 ( x 3 ) + 8 = 0 {\displaystyle (x^{3})^{2}-9(x^{3})+8=0} (this is a simple case of a polynomial decomposition). Thus the equation may be simplified by defining a new variable u = x 3 {\displaystyle u=x^{3}} . ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Change of variables Summary Change_of_variable {\displaystyle u=1\quad {\text{and}}\quad u=8.} The solutions in terms of the original variable are obtained by substituting x3 back in for u, which gives x 3 = 1 and x 3 = 8. {\displaystyle x^{3}=1\quad {\text{and}}\quad x^{3}=8.} Then, assuming that one is interested onl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Chaos machine Summary Chaos_machine In mathematics, a chaos machine is a class of algorithms constructed on the base of chaos theory (mainly deterministic chaos) to produce pseudo-random oracle. It represents the idea of creating a universal scheme with modular design and customizable parameters, which can be applied w... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
List of chaotic maps Summary List_of_chaotic_maps In mathematics, a chaotic map is a map (namely, an evolution function) that exhibits some sort of chaotic behavior. Maps may be parameterized by a discrete-time or a continuous-time parameter. Discrete maps usually take the form of iterated functions. Chaotic maps often... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
List of chaotic maps Summary List_of_chaotic_maps Chaotic maps often generate fractals. Although a fractal may be constructed by an iterative procedure, some fractals are studied in and of themselves, as sets rather than in terms of the map that generates them. This is often because there are several different iterativ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Character group Summary Character_group In mathematics, a character group is the group of representations of a group by complex-valued functions. These functions can be thought of as one-dimensional matrix representations and so are special cases of the group characters that arise in the related context of character th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Character group Summary Character_group The characters of irreducible representations are orthogonal.The primary importance of the character group for finite abelian groups is in number theory, where it is used to construct Dirichlet characters. The character group of the cyclic group also appears in the theory of the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Character (mathematics) Summary Unitary_character In mathematics, a character is (most commonly) a special kind of function from a group to a field (such as the complex numbers). There are at least two distinct, but overlapping meanings. Other uses of the word "character" are almost always qualified. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Pólya-Vinogradov inequality Summary Character_sum In mathematics, a character sum is a sum ∑ χ ( n ) {\textstyle \sum \chi (n)} of values of a Dirichlet character χ modulo N, taken over a given range of values of n. Such sums are basic in a number of questions, for example in the distribution of quadratic residues, and... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Characteristic number Summary Characteristic_numbers In mathematics, a characteristic class is a way of associating to each principal bundle of X a cohomology class of X. The cohomology class measures the extent the bundle is "twisted" and whether it possesses sections. Characteristic classes are global invariants that... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Characterization theorem Summary Characterization_(mathematics) In mathematics, a characterization of an object is a set of conditions that, while different from the definition of the object, is logically equivalent to it. To say that "Property P characterizes object X" is to say that not only does X have property P, b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Characterization theorem Summary Characterization_(mathematics) Common mathematical expressions for a characterization of X in terms of P include "P is necessary and sufficient for X", and "X holds if and only if P". It is also common to find statements such as "Property Q characterizes Y up to isomorphism". The first ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Characterization theorem Summary Characterization_(mathematics) A reference on mathematical terminology notes that characteristic originates from the Greek term kharax, "a pointed stake":From Greek kharax came kharakhter, an instrument used to mark or engrave an object. Once an object was marked, it became distinctive,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Characterization theorem Summary Characterization_(mathematics) Characterization is not unique to mathematics, but since the science is abstract, much of the activity can be described as "characterization". For instance, in Mathematical Reviews, as of 2018, more than 24,000 articles contain the word in the article titl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Chiral algebra Summary Chiral_algebra In mathematics, a chiral algebra is an algebraic structure introduced by Beilinson & Drinfeld (2004) as a rigorous version of the rather vague concept of a chiral algebra in physics. In Chiral Algebras, Beilinson and Drinfeld introduced the notion of chiral algebra, which based on ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Chord diagram (mathematics) Summary Chord_diagram_(mathematics) In mathematics, a chord diagram consists of a cyclic order on a set of objects, together with a one-to-one pairing (perfect matching) of those objects. Chord diagrams are conventionally visualized by arranging the objects in their order around a circle, an... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Chord diagram (mathematics) Summary Chord_diagram_(mathematics) {\displaystyle (2n-1)!!} . There is a Catalan number of chord diagrams on a given ordered set in which no two chords cross each other. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Chord diagram (mathematics) Summary Chord_diagram_(mathematics) The crossing pattern of chords in a chord diagram may be described by a circle graph, the intersection graph of the chords: it has a vertex for each chord and an edge for each two chords that cross.In knot theory, a chord diagram can be used to describe th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Principal circle bundle Summary Principal_circle_bundle In mathematics, a circle bundle is a fiber bundle where the fiber is the circle S 1 {\displaystyle S^{1}} . Oriented circle bundles are also known as principal U(1)-bundles, or equivalently, as principal SO(2)-bundles. In physics, circle bundles are the natural ge... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Class formation Summary Class_formation In mathematics, a class formation is a topological group acting on a module satisfying certain conditions. Class formations were introduced by Emil Artin and John Tate to organize the various Galois groups and modules that appear in class field theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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