text
stringlengths 9
3.55k
| source
stringlengths 31
280
|
|---|---|
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 function. It encodes the common concept of relation: an element x is related to an element y, if and only if the pair (x, y) belongs to the set of ordered pairs that defines the binary relation. A binary relation is the most studied special case n = 2 of an n-ary relation over sets X1, ..., Xn, which is a subset of the Cartesian product X 1 × ⋯ × X n .
|
https://en.wikipedia.org/wiki/Left-unique_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 integer z that is a multiple of p, but not to an integer that is not a multiple of p. In this relation, for instance, the prime number 2 is related to numbers such as −4, 0, 6, 10, but not to 1 or 9, just as the prime number 3 is related to 0, 6, and 9, but not to 4 or 13. Binary relations are used in many branches of mathematics to model a wide variety of concepts.
|
https://en.wikipedia.org/wiki/Left-unique_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 defined as a special kind of binary relation. Binary relations are also heavily used in computer science.
|
https://en.wikipedia.org/wiki/Left-unique_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://en.wikipedia.org/wiki/Left-unique_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 union, intersection, and complementation, and satisfying the laws of an algebra of sets.
|
https://en.wikipedia.org/wiki/Left-unique_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 involves decomposing them into subsets called concepts, and placing them in a complete lattice. In some systems of axiomatic set theory, relations are extended to classes, which are generalizations of sets. This extension is needed for, among other things, modeling the concepts of "is an element of" or "is a subset of" in set theory, without running into logical inconsistencies such as Russell's paradox. The terms correspondence, dyadic relation and two-place relation are synonyms for binary relation, though some authors use the term "binary relation" for any subset of a Cartesian product X × Y {\displaystyle X\times Y} without reference to X and Y, and reserve the term "correspondence" for a binary relation with reference to X and Y.
|
https://en.wikipedia.org/wiki/Left-unique_relation
|
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 another example, "is sister of" is a relation on the set of all people, it holds e.g. between Marie Curie and Bronisława Dłuska, and likewise vice versa. Set members may not be in relation "to a certain degree" - either they are in relation or they are not.
|
https://en.wikipedia.org/wiki/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 natural numbers that contains both (1,3) and (3,4), but neither (3,1) nor (4,4). The relation "is a nontrivial divisor of" on the set of one-digit natural numbers is sufficiently small to be shown here: Rdiv = { (2,4), (2,6), (2,8), (3,6), (3,9), (4,8) }; for example 2 is a nontrivial divisor of 8, but not vice versa, hence (2,8) ∈ Rdiv, but (8,2) ∉ Rdiv. If R is a relation that holds for x and y one often writes xRy.
|
https://en.wikipedia.org/wiki/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; some authors also write "(1,3) ∈ (<)".
|
https://en.wikipedia.org/wiki/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 yRz always implies xRz.
|
https://en.wikipedia.org/wiki/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 not may be a matter of definition (is every woman a sister of herself? ), "is ancestor of" is transitive, while "is parent of" is not.
|
https://en.wikipedia.org/wiki/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 relation that is reflexive, antisymmetric, and transitive, an equivalence relation is a relation that is reflexive, symmetric, and transitive, a function is a relation that is right-unique and left-total (see below).Since relations are sets, they can be manipulated using set operations, including union, intersection, and complementation, and satisfying the laws of an algebra of sets. Beyond that, operations like the converse of a relation and the composition of relations are available, satisfying the laws of a calculus of relations.The above concept of relation has been generalized to admit relations between members of two different sets (heterogeneous relation, like "lies on" between the set of all points and that of all lines in geometry), relations between three or more sets (Finitary relation, like "person x lives in town y at time z"), and relations between classes (like "is an element of" on the class of all sets, see Binary relation § Sets versus classes).
|
https://en.wikipedia.org/wiki/Relation_(mathematics)
|
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 Hall (1969). Elliott (2006) showed that binomial rings are essentially the same as λ-rings for which all Adams operations are the identity.
|
https://en.wikipedia.org/wiki/Binomial_ring
|
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) introduced the concept of polynomials biorthogonal with respect to a sequence of measures, while Szegő introduced the concept of two sequences of polynomials that are biorthogonal with respect to each other.
|
https://en.wikipedia.org/wiki/Biorthogonal_polynomials
|
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,j}} is the Kronecker delta. An example is the pair of sets of respectively left and right eigenvectors of a matrix, indexed by eigenvalue, if the eigenvalues are distinct.A biorthogonal system in which E = F {\displaystyle E=F} and v ~ i = u ~ i {\displaystyle {\tilde {v}}_{i}={\tilde {u}}_{i}} is an orthonormal system.
|
https://en.wikipedia.org/wiki/Biorthogonal_system
|
In mathematics, a bipartite matroid is a matroid all of whose circuits have even size.
|
https://en.wikipedia.org/wiki/Bipartite_matroid
|
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://en.wikipedia.org/wiki/Biquadratic_field
|
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://en.wikipedia.org/wiki/Biquaternion_algebra
|
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. When the chessboard has dimensions m × n {\displaystyle m\times n} , then the induced graph is called the m × n {\displaystyle m\times n} bishop's graph.
|
https://en.wikipedia.org/wiki/Bishop's_graph
|
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&g&h&d\\c&g&i&g&c\\d&h&g&f&b\\e&d&c&b&a\end{bmatrix}}={\begin{bmatrix}a_{11}&a_{12}&a_{13}&a_{14}&a_{15}\\a_{12}&a_{22}&a_{23}&a_{24}&a_{14}\\a_{13}&a_{23}&a_{33}&a_{23}&a_{13}\\a_{14}&a_{24}&a_{23}&a_{22}&a_{12}\\a_{15}&a_{14}&a_{13}&a_{12}&a_{11}\end{bmatrix}}} is bisymmetric. The associated 5 × 5 {\displaystyle 5\times 5} exchange matrix for this example is J 5 = {\displaystyle J_{5}={\begin{bmatrix}0&0&0&0&1\\0&0&0&1&0\\0&0&1&0&0\\0&1&0&0&0\\1&0&0&0&0\end{bmatrix}}}
|
https://en.wikipedia.org/wiki/Bisymmetric_matrix
|
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 by J. C. Kelly in the study of quasimetrics, i.e. distance functions that are not required to be symmetric.
|
https://en.wikipedia.org/wiki/Bitopological_space
|
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 theory. In general, a homology theory is a covariant functor from the category of spaces to the category of abelian groups, while a cohomology theory is a contravariant functor from the category of (nice) spaces to the category of rings. A bivariant theory is a functor both covariant and contravariant; hence, the name “bivariant”.
|
https://en.wikipedia.org/wiki/Bivariant_Chow_group
|
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 . {\displaystyle x=x_{1}+\mathrm {h} x_{2},\ y=y_{1}+\mathrm {h} y_{2},\ z=z_{1}+\mathrm {h} z_{2},\quad \mathrm {h} ^{2}=-1=\mathrm {i} ^{2}=\mathrm {j} ^{2}=\mathrm {k} ^{2}.}
|
https://en.wikipedia.org/wiki/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 = x 1 i + y 1 j + z 1 k {\displaystyle r_{1}=x_{1}\mathrm {i} +y_{1}\mathrm {j} +z_{1}\mathrm {k} } and r 2 = x 2 i + y 2 j + z 2 k {\displaystyle r_{2}=x_{2}\mathrm {i} +y_{2}\mathrm {j} +z_{2}\mathrm {k} } are vectors. Thus the bivector q = x i + y j + z k = r 1 + h r 2 . {\displaystyle q=x\mathrm {i} +y\mathrm {j} +z\mathrm {k} =r_{1}+\mathrm {h} r_{2}.}
|
https://en.wikipedia.org/wiki/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 {x + yr1: x, y ∈ R}. Such a circle corresponds to the space rotation parameters of the Lorentz group.
|
https://en.wikipedia.org/wiki/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. In the words of Ronald Shaw, "Bivectors are logarithms of Lorentz transformations.
|
https://en.wikipedia.org/wiki/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 bivectors under commutation. The Lie algebra of bivectors is essentially that of complex 3-vectors, with the Lie product being defined to be the familiar cross product in (complex) 3-dimensional space.William Rowan Hamilton coined both the terms vector and bivector. The first term was named with quaternions, and the second about a decade later, as in Lectures on Quaternions (1853).
|
https://en.wikipedia.org/wiki/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 biquaternions as 2 × 2 complex matrices acting on the complex plane with basis {1, h}, ( h v w + h x − w + h x − h v ) {\displaystyle {\begin{pmatrix}hv&w+hx\\-w+hx&-hv\end{pmatrix}}} represents bivector q = vi + wj + xk.The conjugate transpose of this matrix corresponds to −q, so the representation of bivector q is a skew-Hermitian matrix.
|
https://en.wikipedia.org/wiki/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 for direction of propagation, one for amplitude." == References ==
|
https://en.wikipedia.org/wiki/Bivector_(complex)
|
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 many areas of mathematics and physics. They are related to complex numbers in two dimensions and to both pseudovectors and quaternions in three dimensions.
|
https://en.wikipedia.org/wiki/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, much as vectors can be thought of as directed line segments.
|
https://en.wikipedia.org/wiki/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 it has the same area, same orientation, and is parallel to the same plane (see figure). Bivectors are generated by the exterior product on vectors: given two vectors a and b, their exterior product a ∧ b is a bivector, as is the sum of any bivectors. Not all bivectors can be generated as a single exterior product. More precisely, a bivector that can be expressed as an exterior product is called simple; in up to three dimensions all bivectors are simple, but in higher dimensions this is not the case. The exterior product of two vectors is alternating, so b ∧ a is the negation of the bivector a ∧ b, producing the opposite orientation, and a ∧ a is the zero bivector.
|
https://en.wikipedia.org/wiki/Bivector
|
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 partition it, into a collection of smaller matrices. Any matrix may be interpreted as a block matrix in one or more ways, with each interpretation defined by how its rows and columns are partitioned.
|
https://en.wikipedia.org/wiki/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 colgroups {\displaystyle {\text{colgroups}}} . The original matrix is then considered as the "total" of these groups, in the sense that the ( i , j ) {\displaystyle (i,j)} entry of the original matrix corresponds in a 1-to-1 way with some ( s , t ) {\displaystyle (s,t)} offset entry of some ( x , y ) {\displaystyle (x,y)} , where x ∈ rowgroups {\displaystyle x\in {\text{rowgroups}}} and y ∈ colgroups {\displaystyle y\in {\text{colgroups}}} . Block matrix algebra arises in general from biproducts in categories of matrices.
|
https://en.wikipedia.org/wiki/Partitioned_matrix
|
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://en.wikipedia.org/wiki/Block_matrix_pseudoinverse
|
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 point. When the context of graph theory is clear, it can be called more simply a bouquet. Although bouquets have a very simple structure as graphs, they are of some importance in topological graph theory because their graph embeddings can still be non-trivial.
|
https://en.wikipedia.org/wiki/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 Cayley–Serre graph (a variant of Cayley graphs with doubled edges) can be represented as the covering graph of a bouquet. == References ==
|
https://en.wikipedia.org/wiki/Bouquet_graph
|
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 where f(a,b;x) is the uniform distribution of x for the interval and H ( x ) {\displaystyle H(x)} is the Heaviside step function. As with most such discontinuous functions, there is a question of the value at the transition points. These values are probably best chosen for each individual application. When a boxcar function is selected as the impulse response of a filter, the result is a simple moving average filter, whose frequency response is a sinc-in-frequency, a type of low-pass filter.
|
https://en.wikipedia.org/wiki/Boxcar_function
|
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 quotient of the algebra Brace{L} obtained by imposing the congruence relations below, where w, w', ..., w" are any monomials in Super: {w} = 0 if length(w) ≠ n {w}{w'}...{w"} = 0 whenever any positive letter a of L occurs more than n times in the monomial {w}{w'}...{w"}. Let {w}{w'}...{w"} be a monomial in Brace{L} in which some positive letter a occurs more than n times, and let b, c, d, e, ..., f, g be any letters in L.
|
https://en.wikipedia.org/wiki/Bracket_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 notion should not be confused with quasitriangular Hopf algebra.
|
https://en.wikipedia.org/wiki/Braided_Hopf_algebra
|
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 fulfilled. Hence drawing tensor diagrams with τ {\displaystyle \tau } an overcrossing the corresponding composed morphism is unchanged when a Reidemeister move is applied to the tensor diagram and thus they present a representation of the braid group. As first example, every vector space is braided via the trivial braiding (simply flipping).
|
https://en.wikipedia.org/wiki/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})=q_{ij}(x_{j}\otimes x_{i})} A good source for braided vector spaces entire braided monoidal categories with braidings between any objects τ V , W {\displaystyle \tau _{V,W}} , most importantly the modules over quasitriangular Hopf algebras and Yetter–Drinfeld modules over finite groups (such as Z 2 {\displaystyle \mathbb {Z} _{2}} above) If V {\displaystyle V} additionally possesses an algebra structure inside the braided category ("braided algebra") one has a braided commutator (e.g. for a superspace the anticommutator): τ := μ ( ( x ⊗ y ) − τ ( x ⊗ y ) ) μ ( x ⊗ y ) := x y {\displaystyle \;_{\tau }:=\mu ((x\otimes y)-\tau (x\otimes y))\qquad \mu (x\otimes y):=xy} Examples of such braided algebras (and even Hopf algebras) are the Nichols algebras, that are by definition generated by a given braided vectorspace. They appear as quantum Borel part of quantum groups and often (e.g. when finite or over an abelian group) possess an arithmetic root system, multiple Dynkin diagrams and a PBW-basis made up of braided commutators just like the ones in semisimple Lie algebras.
|
https://en.wikipedia.org/wiki/Braided_vector_space
|
In mathematics, a branched covering is a map that is almost a covering map, except on a small set.
|
https://en.wikipedia.org/wiki/Ramified_cover
|
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" homeomorphically projecting into the same differentiable n-disk in Rn. Branched manifolds first appeared in the dynamical systems theory, in connection with one-dimensional hyperbolic attractors constructed by Smale and were formalized by R. F. Williams in a series of papers on expanding attractors. Special cases of low dimensions are known as train tracks (n = 1) and branched surfaces (n = 2) and play prominent role in the geometry of three-manifolds after Thurston.
|
https://en.wikipedia.org/wiki/Branched_manifold
|
In mathematics, a branched surface is a generalization of both surfaces and train tracks.
|
https://en.wikipedia.org/wiki/Branched_surface
|
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}} . The exact dimension n {\displaystyle n} of the space of the new time parameter varies from authors. We follow John B. Walsh and define the ( n , d ) {\displaystyle (n,d)} -brownian sheet, while some authors define the brownian sheet specifically only for n = 2 {\displaystyle n=2} , what we call the ( 2 , d ) {\displaystyle (2,d)} -brownian sheet.
|
https://en.wikipedia.org/wiki/Brownian_sheet
|
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 the structure of exceptional groups of Lie type. The more specialized theory of Bruhat–Tits buildings (named also after François Bruhat) plays a role in the study of p-adic Lie groups analogous to that of the theory of symmetric spaces in the theory of Lie groups.
|
https://en.wikipedia.org/wiki/Building_(mathematics)
|
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 = 0, and is therefore a unicursal (rational) curve of genus zero. If f ( z ) = ∑ n = 0 ∞ ( 2 n n ) z 2 n + 1 = z + 2 z 3 + 6 z 5 + 20 z 7 + ⋯ {\displaystyle f(z)=\sum _{n=0}^{\infty }{2n \choose n}z^{2n+1}=z+2z^{3}+6z^{5}+20z^{7}+\cdots } then y = f ( x 2 a ) ± 2 b {\displaystyle y=f\left({\frac {x}{2a}}\right)\pm 2b\ } are the two branches of the bullet curve at the origin.
|
https://en.wikipedia.org/wiki/Bullet-nose_curve
|
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 bump functions with domain R n {\displaystyle \mathbb {R} ^{n}} forms a vector space, denoted C 0 ∞ ( R n ) {\displaystyle \mathrm {C} _{0}^{\infty }(\mathbb {R} ^{n})} or C c ∞ ( R n ) . {\displaystyle \mathrm {C} _{\mathrm {c} }^{\infty }(\mathbb {R} ^{n}).} The dual space of this space endowed with a suitable topology is the space of distributions.
|
https://en.wikipedia.org/wiki/Test_function
|
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://en.wikipedia.org/wiki/Bundle_gerbe
|
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 with E and B sets. It is no longer true that the preimages π − 1 ( x ) {\displaystyle \pi ^{-1}(x)} must all look alike, unlike fiber bundles where the fibers must all be isomorphic (in the case of vector bundles) and homeomorphic.
|
https://en.wikipedia.org/wiki/Bundle_(mathematics)
|
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 which category of fiber bundles is under consideration. In the first three sections, we will consider general fiber bundles in the category of topological spaces. Then in the fourth section, some other examples will be given.
|
https://en.wikipedia.org/wiki/Bundle_homomorphism
|
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. In this case the element being cancelled out is appearing as the left factors of a·b and a·c and hence it is a case of the left cancellation property. The right cancellation property can be defined analogously.
|
https://en.wikipedia.org/wiki/Cancellative_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 embeddable in a group. Moreover, every finite cancellative semigroup is a group. One of the main problems associated with the study of cancellative semigroups is to determine the necessary and sufficient conditions for embedding a cancellative semigroup in a group. The origins of the study of cancellative semigroups can be traced to the first substantial paper on semigroups, (Suschkewitsch 1928).
|
https://en.wikipedia.org/wiki/Cancellative_semigroup
|
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 by the monomials, ( X i ) i {\displaystyle (X^{i})_{i}} . For finite extension fields, it means the polynomial basis. In linear algebra, it refers to a set of n linearly independent generalized eigenvectors of an n×n matrix A {\displaystyle A} , if the set is composed entirely of Jordan chains. In representation theory, it refers to the basis of the quantum groups introduced by Lusztig.
|
https://en.wikipedia.org/wiki/Canonical_basis
|
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 convention). A closely related notion is a structure map or structure morphism; the map or morphism that comes with the given structure on the object.
|
https://en.wikipedia.org/wiki/Canonical_homomorphism
|
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 problem of defining a canonical map see Kevin Buzzard's talk at the 2022 Grothendieck conference.
|
https://en.wikipedia.org/wiki/Canonical_homomorphism
|
In mathematics, a cardinal function (or cardinal invariant) is a function that returns cardinal numbers.
|
https://en.wikipedia.org/wiki/Character_(topology)
|
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 embedding j: V → M from V into a transitive inner model M with critical point κ and V j n ( κ ) {\displaystyle V_{j^{n}(\kappa )}} ⊆ M. Akihiro Kanamori has shown that the consistency strength of an n+1-superstrong cardinal exceeds that of an n-huge cardinal for each n > 0.
|
https://en.wikipedia.org/wiki/Superstrong_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 ) M ⊂ M . {\displaystyle {}^{j(\kappa )}M\subset M.} Here, α M {\displaystyle {}^{\alpha }M} is the class of all sequences of length α {\displaystyle \alpha } whose elements are in M {\displaystyle M} . Huge cardinals were introduced by Kenneth Kunen (1978).
|
https://en.wikipedia.org/wiki/Ω-huge_cardinal
|
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 often denoted with the Hebrew letter ℵ {\displaystyle \aleph } (aleph) marked with subscript indicating their rank among the infinite cardinals. Cardinality is defined in terms of bijective functions.
|
https://en.wikipedia.org/wiki/Cardinal_addition
|
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.
|
https://en.wikipedia.org/wiki/Cardinal_addition
|
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 cardinality as the original set—something that cannot happen with proper subsets of finite sets. There is a transfinite sequence of cardinal numbers: 0 , 1 , 2 , 3 , … , n , … ; ℵ 0 , ℵ 1 , ℵ 2 , … , ℵ α , … .
|
https://en.wikipedia.org/wiki/Cardinal_addition
|
{\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, this transfinite sequence includes every cardinal number.
|
https://en.wikipedia.org/wiki/Cardinal_addition
|
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 analysis. In category theory, the cardinal numbers form a skeleton of the category of sets.
|
https://en.wikipedia.org/wiki/Cardinal_addition
|
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://en.wikipedia.org/wiki/Suslin_cardinal
|
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 are only the identity morphisms.
|
https://en.wikipedia.org/wiki/Categorical_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://en.wikipedia.org/wiki/Categorical_ring
|
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 example is the category of sets, whose objects are sets and whose arrows are functions. Category theory is a branch of mathematics that seeks to generalize all of mathematics in terms of categories, independent of what their objects and arrows represent.
|
https://en.wikipedia.org/wiki/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 mathematics to set theory and other proposed axiomatic foundations. In general, the objects and arrows may be abstract entities of any kind, and the notion of category provides a fundamental and abstract way to describe mathematical entities and their relationships.
|
https://en.wikipedia.org/wiki/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 arrows, and the same associative method of composing any pair of arrows. Two different categories may also be considered "equivalent" for purposes of category theory, even if they do not have precisely the same structure.
|
https://en.wikipedia.org/wiki/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 continuous maps. All of the preceding categories have the identity map as identity arrows and composition as the associative operation on arrows. The classic and still much used text on category theory is Categories for the Working Mathematician by Saunders Mac Lane.
|
https://en.wikipedia.org/wiki/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 represented by the elements of the monoid), and so can any preorder.
|
https://en.wikipedia.org/wiki/Category_(mathematics)
|
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 objects A {\displaystyle A} , the canonical map 0 → A × 0 {\displaystyle 0\to A\times 0} is an isomorphism (where 0 denotes the initial object). Equivalently, if for every object A {\displaystyle A} the endofunctor A × − {\displaystyle A\times -} defined by B ↦ A × B {\displaystyle B\mapsto A\times B} preserves coproducts up to isomorphisms f {\displaystyle f} . It follows that f {\displaystyle f} and aforementioned canonical maps are equal for each choice of objects. In particular, if the functor A × − {\displaystyle A\times -} has a right adjoint (i.e., if the category is cartesian closed), it necessarily preserves all colimits, and thus any cartesian closed category with finite coproducts (i.e., any bicartesian closed category) is distributive.
|
https://en.wikipedia.org/wiki/Distributive_category
|
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 partitioners of most Catholic churches.
|
https://en.wikipedia.org/wiki/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 semigroups are both left and right reductive, that is, their representations by inner left and right translations are faithful. A regular semigroup is both catholic and orthodox if and only if the semigroup is an inverse semigroup.
|
https://en.wikipedia.org/wiki/Catholic_semigroup
|
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 how the images are included in the kernels. A cochain complex is similar to a chain complex, except that its homomorphisms are in the opposite direction. The homology of a cochain complex is called its cohomology.
|
https://en.wikipedia.org/wiki/Cochain_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 complex is called the singular homology of X, and is a commonly used invariant of a topological space. Chain complexes are studied in homological algebra, but are used in several areas of mathematics, including abstract algebra, Galois theory, differential geometry and algebraic geometry. They can be defined more generally in abelian categories.
|
https://en.wikipedia.org/wiki/Cochain_complexes
|
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 operation that is related to substitution. However these are different operations, as can be seen when considering differentiation (chain rule) or integration (integration by substitution).
|
https://en.wikipedia.org/wiki/Change_of_variables
|
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).
|
https://en.wikipedia.org/wiki/Change_of_variables
|
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}}} into the polynomial gives u 2 − 9 u + 8 = 0 , {\displaystyle u^{2}-9u+8=0,} which is just a quadratic equation with the two solutions: u = 1 and u = 8.
|
https://en.wikipedia.org/wiki/Change_of_variables
|
{\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 original equation are x = ( 1 ) 1 / 3 = 1 and x = ( 8 ) 1 / 3 = 2. {\displaystyle x=(1)^{1/3}=1\quad {\text{and}}\quad x=(8)^{1/3}=2.}
|
https://en.wikipedia.org/wiki/Change_of_variables
|
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 is needed.Theoretical model was published in early 2015 by Maciej A. Czyzewski. It was designed specifically to combine the benefits of hash function and pseudo-random function. However, it can be used to implement many cryptographic primitives, including cryptographic hashes, message authentication codes and randomness extractors.
|
https://en.wikipedia.org/wiki/Chaos_machine
|
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.
|
https://en.wikipedia.org/wiki/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 iterative procedures to generate the same fractal.
|
https://en.wikipedia.org/wiki/List_of_chaotic_maps
|
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 matrices, the function defined by the trace of the matrices is called a character; however, these traces do not in general form a group. Some important properties of these one-dimensional characters apply to characters in general: Characters are invariant on conjugacy classes.
|
https://en.wikipedia.org/wiki/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 discrete Fourier transform. For locally compact abelian groups, the character group (with an assumption of continuity) is central to Fourier analysis.
|
https://en.wikipedia.org/wiki/Character_group
|
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://en.wikipedia.org/wiki/Character_(mathematics)
|
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 finding an upper bound for the least quadratic non-residue modulo N. Character sums are often closely linked to exponential sums by the Gauss sums (this is like a finite Mellin transform). Assume χ is a non-principal Dirichlet character to the modulus N.
|
https://en.wikipedia.org/wiki/Pólya-Vinogradov_inequality
|
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 from a global product structure. They are one of the unifying geometric concepts in algebraic topology, differential geometry, and algebraic geometry. The notion of characteristic class arose in 1935 in the work of Eduard Stiefel and Hassler Whitney about vector fields on manifolds.
|
https://en.wikipedia.org/wiki/Characteristic_number
|
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 defining property of X). Similarly, a set of properties P is said to characterize X, when these properties distinguish X from all other objects. Even though a characterization identifies an object in a unique way, several characterizations can exist for a single object.
|
https://en.wikipedia.org/wiki/Characterization_theorem
|
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 P is a singleton set, while the second says that the extension of Q is a single equivalence class (for isomorphism, in the given example — depending on how up to is being used, some other equivalence relation might be involved).
|
https://en.wikipedia.org/wiki/Characterization_theorem
|
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 nature. The Late Greek suffix -istikos converted the noun character into the adjective characteristic, which, in addition to maintaining its adjectival meaning, later became a noun as well.Just as in chemistry, the characteristic property of a material will serve to identify a sample, or in the study of materials, structures and properties will determine characterization, in mathematics there is a continual effort to express properties that will distinguish a desired feature in a theory or system.
|
https://en.wikipedia.org/wiki/Characterization_theorem
|
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 of objects and features, characterizations have been expressed via the heterogeneous relation aRb, meaning that object a has feature b. For example, b may mean abstract or concrete. The objects can be considered the extensions of the world, while the features are expression of the intensions. A continuing program of characterization of various objects leads to their categorization.
|
https://en.wikipedia.org/wiki/Characterization_theorem
|
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-modules. They give an 'coordinate independent' notion of vertex algebras, which are based on formal power series. Chiral algebras on curves are essentially conformal vertex algebras.
|
https://en.wikipedia.org/wiki/Chiral_algebra
|
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 number of different chord diagrams that may be given for a set of 2 n {\displaystyle 2n} cyclically ordered objects is the double factorial ( 2 n − 1 ) ! !
|
https://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/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 the sequence of crossings along the planar projection of a knot, with each point at which a crossing occurs paired with the point that crosses it. To fully describe the knot, the diagram should be annotated with an extra bit of information for each pair, indicating which point crosses over and which crosses under at that crossing. With this extra information, the chord diagram of a knot is called a Gauss diagram. In the Gauss diagram of a knot, every chord crosses an even number of other chords, or equivalently each pair in the diagram connects a point in an even position of the cyclic order with a point in an odd position, and sometimes this is used as a defining condition of Gauss diagrams.In algebraic geometry, chord diagrams can be used to represent the singularities of algebraic plane curves. == References ==
|
https://en.wikipedia.org/wiki/Chord_diagram_(mathematics)
|
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 a special case of a sphere bundle.
|
https://en.wikipedia.org/wiki/Principal_circle_bundle
|
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://en.wikipedia.org/wiki/Class_formation
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.