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c_znwkrbqwy2qa | 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 generalization of the more widely understood idea of a unary ... | Left-unique relation |
c_3hp71aptjddn | {\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 integer z that is a multiple of p, but not to an integ... | Left-unique relation |
c_8shepejwasas | 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 defined as a special kind of binary relation. Binary re... | Left-unique relation |
c_xwgzlwk92vgl | 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 . | Left-unique relation |
c_mou29ralwhf4 | {\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 union, intersection, and complementation, and satisfy... | Left-unique relation |
c_yz6v5bmxfl88 | 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 involves decomposing them into subsets called conce... | Left-unique relation |
c_l6m9wh17q7uw | 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), but neither between 3 and 1 nor between 4 and 4. As ano... | Relation (mathematics) |
c_eyno0nwli2fk | 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 natural numbers that contains both (1,3) and (3,4), but ... | Relation (mathematics) |
c_54sh9hahis4w | 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; some authors also write "(1,3) ∈ (<)". | Relation (mathematics) |
c_1ch71raoo69c | 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 yRz always implies xRz. | Relation (mathematics) |
c_xbwmxi1uaoyk | 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 not may be a matter of definition (is every woman a s... | Relation (mathematics) |
c_pv9stmavlqza | 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 relation that is reflexive, antisymmetric, and transit... | Relation (mathematics) |
c_uwbplp355sm2 | 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. Binomial rings were introduced by Hal... | Binomial ring |
c_xrd3zs7vmb1o | 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 biorthogonal polynomials in the literature: Iserles & Nørsett (1988)... | Biorthogonal polynomials |
c_7moz2ritzfco | 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 bilinear mapping and δ i , j {\displaystyle \delta _{i... | Biorthogonal system |
c_5os95i8j30ms | In mathematics, a bipartite matroid is a matroid all of whose circuits have even size. | Bipartite matroid |
c_wrjil78oe8u6 | 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. | Biquadratic field |
c_osewrkrpu751 | 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. | Biquaternion algebra |
c_0jeboqoekt6u | 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) if they occupy a common diagonal. Whe... | Bishop's graph |
c_4o7gz3jszu2m | 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 = {\displaystyle {\begin{bmatrix}a&b&c&d&e\\b&f... | Bisymmetric matrix |
c_2ce1fr7igyk0 | 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 ) {\displaystyle (X,\sigma ,\tau )} . The notion was introduced... | Bitopological space |
c_ts1cfww03c9o | 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 mix of a homology theory and a cohomology theo... | Bivariant Chow group |
c_684hahxr3mj9 | 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 , z = z 1 + h z 2 , h 2 = − 1 = i 2 = j 2 = k 2 ... | Bivector (complex) |
c_mo67lorlfdex | 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 = x 1 i + y 1 j + z 1 k {\displaystyle r_{1}=x_{... | Bivector (complex) |
c_b58swicn0fn0 | 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 {x + yr1: x, y ∈ R}. Such a circle corresponds... | Bivector (complex) |
c_i02ap6bkooh9 | 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. In the words of Ronald Shaw, "Bivectors are l... | Bivector (complex) |
c_1ckft9yl38q9 | "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 bivectors under commutation. The Lie algebra of bivect... | Bivector (complex) |
c_fkh3a2i2jc38 | : 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 biquaternions as 2 × 2 complex matrices acting on th... | Bivector (complex) |
c_faazrnewumo9 | 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 for direction of propagation, one for amplitud... | Bivector (complex) |
c_oc1w1glkaqmw | 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. Bivectors have applications in ma... | Bivector |
c_uub4qxrbfa0l | 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, much as vectors can be tho... | Bivector |
c_zoj3nla03jp6 | 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 it has the same area, same... | Bivector |
c_vyunh033bhjc | 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 horizontal and vertical lines, which break it up, or ... | Partitioned matrix |
c_60aia6us749o | 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 colgroups {\displaystyle {\text{colgroups}}} ... | Partitioned matrix |
c_x46if05ddatd | 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. | Block matrix pseudoinverse |
c_q1p5og8393yl | 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 {\displaystyle m} circles joined at a ... | Bouquet graph |
c_cscl6phndpt2 | 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 Cayley–Serre graph (a variant of Cayley g... | Bouquet graph |
c_ng11euuxt8lf | 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 in terms of the uniform distribution as... | Boxcar function |
c_2830bho0yw6y | 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 dimension n over the field K is the quoti... | Bracket algebra |
c_euxp1mwrl2cg | 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 notion should not be confused with quasitriangula... | Braided Hopf algebra |
c_4ss682abu7jg | 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\otimes V} such that the Yang–Baxter equation is fulf... | Braided vector space |
c_8z2c4eef9amf | 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})=q_{ij}(x_{j}\otimes x_{i})} A good source for brai... | Braided vector space |
c_7h0k5chwbvgy | In mathematics, a branched covering is a map that is almost a covering map, except on a small set. | Ramified cover |
c_vqr9gtl1nkuk | 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 which involves one or several "branches" homeomo... | Branched manifold |
c_fhf520zaym7t | In mathematics, a branched surface is a generalization of both surfaces and train tracks. | Branched surface |
c_8w6i0c63sfon | 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 + n {\displaystyle \mathbb {R} _{+}^{n}}... | Brownian sheet |
c_pw8h2rkwjx25 | 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 initially introduced by Jacques Tits as a means to understand... | Building (mathematics) |
c_0xv11gayxtw6 | 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 and y = 0, x = 0 and z = 0, and y = 0 and z ... | Bullet-nose curve |
c_kpvcuvxv8g9q | 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 compactly supported. The set of all b... | Test function |
c_antogm2r565j | In mathematics, a bundle gerbe is a geometrical model of certain 1-gerbes with connection, or equivalently of a 2-class in Deligne cohomology. | Bundle gerbe |
c_kvxe0b1zlzsm | 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 bundles. For example, one can consider a bundle π: E→ B... | Bundle (mathematics) |
c_4qqse4uqitd0 | 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 variations on the basic theme, depending on precisely ... | Bundle homomorphism |
c_8rpnd1w7j77i | 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 cancel the element a and deduce the equality b = c. I... | Cancellative semigroup |
c_irnk54gkpm1r | 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 embeddable in a group. Moreover, every finite cancellative ... | Cancellative semigroup |
c_rt4nupix9h3v | 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 ring, it refers to its standard basis given... | Canonical basis |
c_15nllm70jj7q | 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 sometimes depends on a convention (e.g., a sign co... | Canonical homomorphism |
c_z54u3wueed39 | 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 example, see prestack. For a discussion of the prob... | Canonical homomorphism |
c_73njiavfp6c1 | In mathematics, a cardinal function (or cardinal invariant) is a function that returns cardinal numbers. | Character (topology) |
c_qqv3led03vm2 | 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 if and only if there exists an elementary embeddi... | Superstrong cardinal |
c_kzxq03ipj104 | 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 ) M ⊂ M . {\displaystyle {}^{j(\kappa )}... | Ω-huge cardinal |
c_7a4nuk1aqjcw | 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 cardinal numbers have been introduced, which are ... | Cardinal addition |
c_q6zrgtsm2i85 | 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 is more complex. | Cardinal addition |
c_2t9idpu21w6b | 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 proper subset of an infinite set to have the same c... | Cardinal addition |
c_f9x2uun3mblv | {\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 ordinal numbers. If the axiom of choice is true, t... | Cardinal addition |
c_5n6can4c9ik8 | 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, combinatorics, abstract algebra and mathematical anal... | Cardinal addition |
c_vqj52rd7f5pn | 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). | Suslin cardinal |
c_okrdv3163oyo | 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 the set R and whose morphisms a... | Categorical ring |
c_fwfbqx1yt0dy | 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. | Categorical ring |
c_7knn3c4dh4kh | 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 existence of an identity arrow for each object. A simple ex... | Category (mathematics) |
c_l0e093nsrprh | 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 mathematics to set theory and other proposed axiomatic foundat... | Category (mathematics) |
c_rky2a8lzynn2 | 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 arrows, and the same associative method of composing a... | Category (mathematics) |
c_qny7sf2hoys8 | 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 continuous maps. All of the preceding categories have the id... | Category (mathematics) |
c_8scx8bwrww8h | 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 represented by the elements of the monoid), and so can an... | Category (mathematics) |
c_h4vq1kbodeu3 | 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\!\times \! (B+C)} is an isomorphism, and for all obj... | Distributive category |
c_vw78mxjgmlb0 | 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 element that is not regular, much like the partiti... | Catholic semigroup |
c_vz1tnr5qi4iu | 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 semigroups are both left and right reductive, that is, ... | Catholic semigroup |
c_3zj5n4gk83ei | 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 to a chain complex is its homology, which describes h... | Cochain complexes |
c_n5ullkfnthtg | 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 complex is called the singular homology of X, and is ... | Cochain complexes |
c_aoqzriwwwk0b | 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 better understood problem. Change of variables is an... | Change of variables |
c_75blam7qzlcv | 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 (see Abel–Ruffini theorem). | Change of variables |
c_f2p74o270zzi | 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}} . Substituting x by u 3 {\displaystyle {\sqrt{u}}... | Change of variables |
c_y4ue66sfpj75 | {\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 only in real solutions, the solutions of the origi... | Change of variables |
c_jkof2ocj422i | 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 wherever randomness and sensitiveness... | Chaos machine |
c_fo5t26gxz1lc | 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 occur in the study of dynamical systems. | List of chaotic maps |
c_3qmc6sx623hw | 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 iterative procedures to generate the same fractal. | List of chaotic maps |
c_lvhk55d448xi | 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 theory. Whenever a group is represented by... | Character group |
c_7sfi61ktp2wm | 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 discrete Fourier transform. For locally ... | Character group |
c_62l38j5wqb5s | 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. | Character (mathematics) |
c_qcykkypah7gg | 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 in particular in the classical question of findin... | Pólya-Vinogradov inequality |
c_6x9ra0i5it1d | 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 measure the deviation of a local product structure f... | Characteristic number |
c_xoa9mtbne2y5 | 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, but that X is the only thing that has property P (i.e., P is a de... | Characterization theorem |
c_k4wyp9qh2epj | 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 type of statement says in different words that the extension of ... | Characterization theorem |
c_tcde6dectfjy | 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, so the character of something came to mean its distinctive natu... | Characterization theorem |
c_s0r8icfdweyw | 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 title, and 93,600 somewhere in the review. In an arbitrary context o... | Characterization theorem |
c_m90om1p2l5eq | 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 the pseudo-tensor category of D-module... | Chiral algebra |
c_s0y6ebdol0my | 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, and drawing the pairs of the matching as chords of the circle. The... | Chord diagram (mathematics) |
c_aw076jyzoss7 | {\displaystyle (2n-1)!!} . There is a Catalan number of chord diagrams on a given ordered set in which no two chords cross each other. | Chord diagram (mathematics) |
c_i09abi6jof6t | 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 the sequence of crossings along the planar projection of a knot, w... | Chord diagram (mathematics) |
c_zsm7i9lu03p9 | 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 geometric setting for electromagnetism. A circle bundle is... | Principal circle bundle |
c_3xg64yot3w9i | 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. | Class formation |
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