title stringlengths 1 113 | text stringlengths 9 3.55k | source stringclasses 1 value |
|---|---|---|
cartan algebra | In a finite-dimensional semisimple Lie algebra over an algebraically closed field of characteristic zero (e.g., C {\displaystyle \mathbb {C} } ), a Cartan subalgebra is the same thing as a maximal abelian subalgebra consisting of elements x such that the adjoint endomorphism ad ( x ): g → g {\displaystyle \operatorname {ad} (x):{\mathfrak {g}}\to {\mathfrak {g}}} is semisimple (i.e., diagonalizable). Sometimes this characterization is simply taken as the definition of a Cartan subalgebra.pg 231In general, a subalgebra is called toral if it consists of semisimple elements. Over an algebraically closed field, a toral subalgebra is automatically abelian. Thus, over an algebraically closed field of characteristic zero, a Cartan subalgebra can also be defined as a maximal toral subalgebra. Kac–Moody algebras and generalized Kac–Moody algebras also have subalgebras that play the same role as the Cartan subalgebras of semisimple Lie algebras (over a field of characteristic zero). | wikipedia |
cauchy-continuous function | In mathematics, a Cauchy-continuous, or Cauchy-regular, function is a special kind of continuous function between metric spaces (or more general spaces). Cauchy-continuous functions have the useful property that they can always be (uniquely) extended to the Cauchy completion of their domain. | wikipedia |
clifford multiplication | In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and digital image processing. They are named after the English mathematician William Kingdon Clifford (1845–1879). The most familiar Clifford algebras, the orthogonal Clifford algebras, are also referred to as (pseudo-)Riemannian Clifford algebras, as distinct from symplectic Clifford algebras. | wikipedia |
colombeau algebra | In mathematics, a Colombeau algebra is an algebra of a certain kind containing the space of Schwartz distributions. While in classical distribution theory a general multiplication of distributions is not possible, Colombeau algebras provide a rigorous framework for this. Such a multiplication of distributions has long been believed to be impossible because of L. Schwartz' impossibility result, which basically states that there cannot be a differential algebra containing the space of distributions and preserving the product of continuous functions. However, if one only wants to preserve the product of smooth functions instead such a construction becomes possible, as demonstrated first by Colombeau. | wikipedia |
colombeau algebra | As a mathematical tool, Colombeau algebras can be said to combine a treatment of singularities, differentiation and nonlinear operations in one framework, lifting the limitations of distribution theory. These algebras have found numerous applications in the fields of partial differential equations, geophysics, microlocal analysis and general relativity so far. Colombeau algebras are named after French mathematician Jean François Colombeau. | wikipedia |
coulomb functions | In mathematics, a Coulomb wave function is a solution of the Coulomb wave equation, named after Charles-Augustin de Coulomb. They are used to describe the behavior of charged particles in a Coulomb potential and can be written in terms of confluent hypergeometric functions or Whittaker functions of imaginary argument. | wikipedia |
kleene algebra (with involution) | In mathematics, a De Morgan algebra (named after Augustus De Morgan, a British mathematician and logician) is a structure A = (A, ∨, ∧, 0, 1, ¬) such that: (A, ∨, ∧, 0, 1) is a bounded distributive lattice, and ¬ is a De Morgan involution: ¬(x ∧ y) = ¬x ∨ ¬y and ¬¬x = x. (i.e. an involution that additionally satisfies De Morgan's laws)In a De Morgan algebra, the laws ¬x ∨ x = 1 (law of the excluded middle), and ¬x ∧ x = 0 (law of noncontradiction)do not always hold. In the presence of the De Morgan laws, either law implies the other, and an algebra which satisfies them becomes a Boolean algebra. Remark: It follows that ¬(x ∨ y) = ¬x ∧ ¬y, ¬1 = 0 and ¬0 = 1 (e.g. ¬1 = ¬1 ∨ 0 = ¬1 ∨ ¬¬0 = ¬(1 ∧ ¬0) = ¬¬0 = 0). | wikipedia |
kleene algebra (with involution) | Thus ¬ is a dual automorphism of (A, ∨, ∧, 0, 1). If the lattice is defined in terms of the order instead, i.e. (A, ≤) is a bounded partial order with a least upper bound and greatest lower bound for every pair of elements, and the meet and join operations so defined satisfy the distributive law, then the complementation can also be defined as an involutive anti-automorphism, that is, a structure A = (A, ≤, ¬) such that: (A, ≤) is a bounded distributive lattice, and ¬¬x = x, and x ≤ y → ¬y ≤ ¬x.De Morgan algebras were introduced by Grigore Moisil around 1935, although without the restriction of having a 0 and a 1. They were then variously called quasi-boolean algebras in the Polish school, e.g. by Rasiowa and also distributive i-lattices by J. A. Kalman. | wikipedia |
kleene algebra (with involution) | (i-lattice being an abbreviation for lattice with involution.) They have been further studied in the Argentinian algebraic logic school of Antonio Monteiro.De Morgan algebras are important for the study of the mathematical aspects of fuzzy logic. The standard fuzzy algebra F = (, max(x, y), min(x, y), 0, 1, 1 − x) is an example of a De Morgan algebra where the laws of excluded middle and noncontradiction do not hold. Another example is Dunn's four-valued semantics for De Morgan algebra, which has the values T(rue), F(alse), B(oth), and N(either), where F < B < T, F < N < T, and B and N are not comparable. | wikipedia |
dirichlet l-function | In mathematics, a Dirichlet L-series is a function of the form L ( s , χ ) = ∑ n = 1 ∞ χ ( n ) n s . {\displaystyle L(s,\chi )=\sum _{n=1}^{\infty }{\frac {\chi (n)}{n^{s}}}.} where χ {\displaystyle \chi } is a Dirichlet character and s a complex variable with real part greater than 1. | wikipedia |
dirichlet l-function | It is a special case of a Dirichlet series. By analytic continuation, it can be extended to a meromorphic function on the whole complex plane, and is then called a Dirichlet L-function and also denoted L(s, χ). These functions are named after Peter Gustav Lejeune Dirichlet who introduced them in (Dirichlet 1837) to prove the theorem on primes in arithmetic progressions that also bears his name. | wikipedia |
dirichlet l-function | In the course of the proof, Dirichlet shows that L(s, χ) is non-zero at s = 1. Moreover, if χ is principal, then the corresponding Dirichlet L-function has a simple pole at s = 1. Otherwise, the L-function is entire. | wikipedia |
dirichlet algebra | In mathematics, a Dirichlet algebra is a particular type of algebra associated to a compact Hausdorff space X. It is a closed subalgebra of C(X), the uniform algebra of bounded continuous functions on X, whose real parts are dense in the algebra of bounded continuous real functions on X. The concept was introduced by Andrew Gleason (1957). | wikipedia |
generalized clifford algebra | In mathematics, a Generalized Clifford algebra (GCA) is a unital associative algebra that generalizes the Clifford algebra, and goes back to the work of Hermann Weyl, who utilized and formalized these clock-and-shift operators introduced by J. J. Sylvester (1882), and organized by Cartan (1898) and Schwinger.Clock and shift matrices find routine applications in numerous areas of mathematical physics, providing the cornerstone of quantum mechanical dynamics in finite-dimensional vector spaces. The concept of a spinor can further be linked to these algebras.The term Generalized Clifford Algebras can also refer to associative algebras that are constructed using forms of higher degree instead of quadratic forms. | wikipedia |
grassmann–cayley algebra | In mathematics, a Grassmann–Cayley algebra is the exterior algebra with an additional product, which may be called the shuffle product or the regressive product. It is the most general structure in which projective properties are expressed in a coordinate-free way. The technique is based on work by German mathematician Hermann Grassmann on exterior algebra, and subsequently by British mathematician Arthur Cayley's work on matrices and linear algebra. It is a form of modeling algebra for use in projective geometry.The technique uses subspaces as basic elements of computation, a formalism which allows the translation of synthetic geometric statements into invariant algebraic statements. This can create a useful framework for the modeling of conics and quadrics among other forms, and in tensor mathematics. It also has a number of applications in robotics, particularly for the kinematical analysis of manipulators. | wikipedia |
green’s function | In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if L {\displaystyle \operatorname {L} } is the linear differential operator, then the Green's function G {\displaystyle G} is the solution of the equation L G = δ {\displaystyle \operatorname {L} G=\delta } , where δ {\displaystyle \delta } is Dirac's delta function; the solution of the initial-value problem L y = f {\displaystyle \operatorname {L} y=f} is the convolution ( G ∗ f {\displaystyle G\ast f} ).Through the superposition principle, given a linear ordinary differential equation (ODE), L y = f {\displaystyle \operatorname {L} y=f} , one can first solve L G = δ s {\displaystyle \operatorname {L} G=\delta _{s}} , for each s, and realizing that, since the source is a sum of delta functions, the solution is a sum of Green's functions as well, by linearity of L. Green's functions are named after the British mathematician George Green, who first developed the concept in the 1820s. In the modern study of linear partial differential equations, Green's functions are studied largely from the point of view of fundamental solutions instead. Under many-body theory, the term is also used in physics, specifically in quantum field theory, aerodynamics, aeroacoustics, electrodynamics, seismology and statistical field theory, to refer to various types of correlation functions, even those that do not fit the mathematical definition. In quantum field theory, Green's functions take the roles of propagators. | wikipedia |
data encoding | In mathematics, a Gödel code was the basis for the proof of Gödel's incompleteness theorem. Here, the idea was to map mathematical notation to a natural number (using a Gödel numbering). | wikipedia |
hecke algebra (disambiguation) | In mathematics, a Hecke algebra is classically the algebra of Hecke operators studied by Erich Hecke. It may also refer to one of several algebras (some of which are related to the classical Hecke algebra): Iwahori–Hecke algebra of a Coxeter group. Hecke algebra of a pair (g,K) where g is the Lie algebra of a Lie group G and K is a compact subgroup of G. Hecke algebra of a locally compact group H(G,K), for a locally compact group G with respect to a compact subgroup K. Hecke algebra of a finite group, the algebra spanned by the double cosets HgH of a subgroup H of a finite group G. Spherical Hecke algebra, when K is a maximal open compact subgroup of a general linear group. Affine Hecke algebra Parabolic Hecke algebra Parahoric Hecke algebra | wikipedia |
hecke algebra of a locally compact group | In mathematics, a Hecke algebra of a locally compact group is an algebra of bi-invariant measures under convolution. | wikipedia |
heyting algebra | In mathematics, a Heyting algebra (also known as pseudo-Boolean algebra) is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped with a binary operation a → b of implication such that (c ∧ a) ≤ b is equivalent to c ≤ (a → b). From a logical standpoint, A → B is by this definition the weakest proposition for which modus ponens, the inference rule A → B, A ⊢ B, is sound. Like Boolean algebras, Heyting algebras form a variety axiomatizable with finitely many equations. Heyting algebras were introduced by Arend Heyting (1930) to formalize intuitionistic logic. | wikipedia |
heyting algebra | As lattices, Heyting algebras are distributive. Every Boolean algebra is a Heyting algebra when a → b is defined as ¬a ∨ b, as is every complete distributive lattice satisfying a one-sided infinite distributive law when a → b is taken to be the supremum of the set of all c for which c ∧ a ≤ b. In the finite case, every nonempty distributive lattice, in particular every nonempty finite chain, is automatically complete and completely distributive, and hence a Heyting algebra. It follows from the definition that 1 ≤ 0 → a, corresponding to the intuition that any proposition a is implied by a contradiction 0. | wikipedia |
heyting algebra | Although the negation operation ¬a is not part of the definition, it is definable as a → 0. The intuitive content of ¬a is the proposition that to assume a would lead to a contradiction. The definition implies that a ∧ ¬a = 0. | wikipedia |
heyting algebra | It can further be shown that a ≤ ¬¬a, although the converse, ¬¬a ≤ a, is not true in general, that is, double negation elimination does not hold in general in a Heyting algebra. Heyting algebras generalize Boolean algebras in the sense that Boolean algebras are precisely the Heyting algebras satisfying a ∨ ¬a = 1 (excluded middle), equivalently ¬¬a = a. Those elements of a Heyting algebra H of the form ¬a comprise a Boolean lattice, but in general this is not a subalgebra of H (see below). Heyting algebras serve as the algebraic models of propositional intuitionistic logic in the same way Boolean algebras model propositional classical logic. | wikipedia |
heyting algebra | The internal logic of an elementary topos is based on the Heyting algebra of subobjects of the terminal object 1 ordered by inclusion, equivalently the morphisms from 1 to the subobject classifier Ω. The open sets of any topological space form a complete Heyting algebra. Complete Heyting algebras thus become a central object of study in pointless topology. Every Heyting algebra whose set of non-greatest elements has a greatest element (and forms another Heyting algebra) is subdirectly irreducible, whence every Heyting algebra can be made subdirectly irreducible by adjoining a new greatest element. | wikipedia |
heyting algebra | It follows that even among the finite Heyting algebras there exist infinitely many that are subdirectly irreducible, no two of which have the same equational theory. Hence no finite set of finite Heyting algebras can supply all the counterexamples to non-laws of Heyting algebra. This is in sharp contrast to Boolean algebras, whose only subdirectly irreducible one is the two-element one, which on its own therefore suffices for all counterexamples to non-laws of Boolean algebra, the basis for the simple truth table decision method. Nevertheless, it is decidable whether an equation holds of all Heyting algebras.Heyting algebras are less often called pseudo-Boolean algebras, or even Brouwer lattices, although the latter term may denote the dual definition, or have a slightly more general meaning. | wikipedia |
hodge algebra | In mathematics, a Hodge algebra or algebra with straightening law is a commutative algebra that is a free module over some ring R, together with a given basis similar to the basis of standard monomials of the coordinate ring of a Grassmannian. Hodge algebras were introduced by Corrado De Concini, David Eisenbud, and Claudio Procesi (1982), who named them after W. V. D. Hodge. | wikipedia |
quasitriangular hopf algebra | Also as a consequence of the properties of quasitriangularity, ( ϵ ⊗ 1 ) R = ( 1 ⊗ ϵ ) R = 1 ∈ H {\displaystyle (\epsilon \otimes 1)R=(1\otimes \epsilon )R=1\in H} ; moreover R − 1 = ( S ⊗ 1 ) ( R ) {\displaystyle R^{-1}=(S\otimes 1)(R)} , R = ( 1 ⊗ S ) ( R − 1 ) {\displaystyle R=(1\otimes S)(R^{-1})} , and ( S ⊗ S ) ( R ) = R {\displaystyle (S\otimes S)(R)=R} . One may further show that the antipode S must be a linear isomorphism, and thus S2 is an automorphism. In fact, S2 is given by conjugating by an invertible element: S 2 ( x ) = u x u − 1 {\displaystyle S^{2}(x)=uxu^{-1}} where u := m ( S ⊗ 1 ) R 21 {\displaystyle u:=m(S\otimes 1)R^{21}} (cf. Ribbon Hopf algebras). It is possible to construct a quasitriangular Hopf algebra from a Hopf algebra and its dual, using the Drinfeld quantum double construction. If the Hopf algebra H is quasitriangular, then the category of modules over H is braided with braiding c U , V ( u ⊗ v ) = T ( R ⋅ ( u ⊗ v ) ) = T ( R 1 u ⊗ R 2 v ) {\displaystyle c_{U,V}(u\otimes v)=T\left(R\cdot (u\otimes v)\right)=T\left(R_{1}u\otimes R_{2}v\right)} . | wikipedia |
hopf algebra | In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an (unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property. The representation theory of a Hopf algebra is particularly nice, since the existence of compatible comultiplication, counit, and antipode allows for the construction of tensor products of representations, trivial representations, and dual representations. Hopf algebras occur naturally in algebraic topology, where they originated and are related to the H-space concept, in group scheme theory, in group theory (via the concept of a group ring), and in numerous other places, making them probably the most familiar type of bialgebra. Hopf algebras are also studied in their own right, with much work on specific classes of examples on the one hand and classification problems on the other. They have diverse applications ranging from condensed-matter physics and quantum field theory to string theory and LHC phenomenology. | wikipedia |
jackson q-bessel function | In mathematics, a Jackson q-Bessel function (or basic Bessel function) is one of the three q-analogs of the Bessel function introduced by Jackson (1906a, 1906b, 1905a, 1905b). The third Jackson q-Bessel function is the same as the Hahn–Exton q-Bessel function. | wikipedia |
jónsson–tarski algebra | In mathematics, a Jónsson–Tarski algebra or Cantor algebra is an algebraic structure encoding a bijection from an infinite set X onto the product X×X. They were introduced by Bjarni Jónsson and Alfred Tarski (1961, Theorem 5). Smirnov (1971), named them after Georg Cantor because of Cantor's pairing function and Cantor's theorem that an infinite set X has the same number of elements as X×X. The term Cantor algebra is also occasionally used to mean the Boolean algebra of all clopen subsets of the Cantor set, or the Boolean algebra of Borel subsets of the reals modulo meager sets (sometimes called the Cohen algebra). The group of order-preserving automorphisms of the free Jónsson–Tarski algebra on one generator is the Thompson group F. | wikipedia |
kac algebra | In mathematics, a Kac–Moody algebra (named for Victor Kac and Robert Moody, who independently and simultaneously discovered them in 1968) is a Lie algebra, usually infinite-dimensional, that can be defined by generators and relations through a generalized Cartan matrix. These algebras form a generalization of finite-dimensional semisimple Lie algebras, and many properties related to the structure of a Lie algebra such as its root system, irreducible representations, and connection to flag manifolds have natural analogues in the Kac–Moody setting. A class of Kac–Moody algebras called affine Lie algebras is of particular importance in mathematics and theoretical physics, especially two-dimensional conformal field theory and the theory of exactly solvable models. Kac discovered an elegant proof of certain combinatorial identities, the Macdonald identities, which is based on the representation theory of affine Kac–Moody algebras. Howard Garland and James Lepowsky demonstrated that Rogers–Ramanujan identities can be derived in a similar fashion. | wikipedia |
regular algebra | In mathematics, a Kleene algebra ( KLAY-nee; named after Stephen Cole Kleene) is an idempotent (and thus partially ordered) semiring endowed with a closure operator. It generalizes the operations known from regular expressions. | wikipedia |
klein geometry | In mathematics, a Klein geometry is a type of geometry motivated by Felix Klein in his influential Erlangen program. More specifically, it is a homogeneous space X together with a transitive action on X by a Lie group G, which acts as the symmetry group of the geometry. For background and motivation see the article on the Erlangen program. | wikipedia |
leavitt path algebra | In mathematics, a Leavitt path algebra is a universal algebra constructed from a directed graph. Leavitt path algebras generalize Leavitt algebras and may be considered as algebraic analogues of graph C*-algebras. | wikipedia |
center of a lie algebra | In mathematics, a Lie algebra (pronounced LEE) is a vector space g {\displaystyle {\mathfrak {g}}} together with an operation called the Lie bracket, an alternating bilinear map g × g → g {\displaystyle {\mathfrak {g}}\times {\mathfrak {g}}\rightarrow {\mathfrak {g}}} , that satisfies the Jacobi identity. Otherwise said, a Lie algebra is an algebra over a field where the multiplication operation is now called Lie bracket and has two additional properties: it is alternating and satisfies the Jacobi identity. The Lie bracket of two vectors x {\displaystyle x} and y {\displaystyle y} is denoted {\displaystyle } . The Lie bracket does not need to be associative, meaning that the Lie algebra can be non associative. | wikipedia |
center of a lie algebra | Given an associative algebra (like for example the space of square matrices), a Lie bracket can be and is often defined through the commutator, namely defining = x y − y x {\displaystyle =xy-yx} correctly defines a Lie bracket in addition to the already existing multiplication operation. Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: any Lie group gives rise to a Lie algebra, which is its tangent space at the identity. Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to finite coverings (Lie's third theorem). | wikipedia |
center of a lie algebra | This correspondence allows one to study the structure and classification of Lie groups in terms of Lie algebras. In physics, Lie groups appear as symmetry groups of physical systems, and their Lie algebras (tangent vectors near the identity) may be thought of as infinitesimal symmetry motions. | wikipedia |
center of a lie algebra | Thus Lie algebras and their representations are used extensively in physics, notably in quantum mechanics and particle physics. An elementary example (that is not derived from an associative algebra) is the space of three dimensional vectors g = R 3 {\displaystyle {\mathfrak {g}}=\mathbb {R} ^{3}} with the Lie bracket operation defined by the cross product = x × y . {\displaystyle =x\times y.} | wikipedia |
center of a lie algebra | This is skew-symmetric since x × y = − y × x {\displaystyle x\times y=-y\times x} , and instead of associativity it satisfies the Jacobi identity: x × ( y × z ) = ( x × y ) × z + y × ( x × z ) . {\displaystyle x\times (y\times z)\ =\ (x\times y)\times z\ +\ y\times (x\times z).} This is the Lie algebra of the Lie group of rotations of space, and each vector v ∈ R 3 {\displaystyle v\in \mathbb {R} ^{3}} may be pictured as an infinitesimal rotation around the axis v {\displaystyle v} , with velocity equal to the magnitude of v {\displaystyle v} . The Lie bracket is a measure of the non-commutativity between two rotations: since a rotation commutes with itself, we have the alternating property = x × x = 0 {\displaystyle =x\times x=0} . | wikipedia |
nilpotent lie algebra | In mathematics, a Lie algebra g {\displaystyle {\mathfrak {g}}} is nilpotent if its lower central series terminates in the zero subalgebra. The lower central series is the sequence of subalgebras g ≥ ≥ ] ≥ ] ] ≥ . . | wikipedia |
nilpotent lie algebra | . {\displaystyle {\mathfrak {g}}\geq \geq ]\geq ]]\geq ...} We write g 0 = g {\displaystyle {\mathfrak {g}}_{0}={\mathfrak {g}}} , and g n = {\displaystyle {\mathfrak {g}}_{n}=} for all n > 0 {\displaystyle n>0} . If the lower central series eventually arrives at the zero subalgebra, then the Lie algebra is called nilpotent. | wikipedia |
nilpotent lie algebra | The lower central series for Lie algebras is analogous to the lower central series in group theory, and nilpotent Lie algebras are analogs of nilpotent groups. The nilpotent Lie algebras are precisely those that can be obtained from abelian Lie algebras, by successive central extensions. Note that the definition means that, viewed as a non-associative non-unital algebra, a Lie algebra g {\displaystyle {\mathfrak {g}}} is nilpotent if it is nilpotent as an ideal. | wikipedia |
derived lie algebra | In mathematics, a Lie algebra g {\displaystyle {\mathfrak {g}}} is solvable if its derived series terminates in the zero subalgebra. The derived Lie algebra of the Lie algebra g {\displaystyle {\mathfrak {g}}} is the subalgebra of g {\displaystyle {\mathfrak {g}}} , denoted {\displaystyle } that consists of all linear combinations of Lie brackets of pairs of elements of g {\displaystyle {\mathfrak {g}}} . The derived series is the sequence of subalgebras g ≥ ≥ , ] ≥ , ] , , ] ] ≥ . . | wikipedia |
derived lie algebra | . {\displaystyle {\mathfrak {g}}\geq \geq ,]\geq ,],,]]\geq ...} If the derived series eventually arrives at the zero subalgebra, then the Lie algebra is called solvable. The derived series for Lie algebras is analogous to the derived series for commutator subgroups in group theory, and solvable Lie algebras are analogs of solvable groups. | wikipedia |
derived lie algebra | Any nilpotent Lie algebra is a fortiori solvable but the converse is not true. The solvable Lie algebras and the semisimple Lie algebras form two large and generally complementary classes, as is shown by the Levi decomposition. The solvable Lie algebras are precisely those that can be obtained from semidirect products, starting from 0 and adding one dimension at a time.A maximal solvable subalgebra is called a Borel subalgebra. The largest solvable ideal of a Lie algebra is called the radical. | wikipedia |
reductive lie algebra | In mathematics, a Lie algebra is reductive if its adjoint representation is completely reducible, hence the name. More concretely, a Lie algebra is reductive if it is a direct sum of a semisimple Lie algebra and an abelian Lie algebra: g = s ⊕ a ; {\displaystyle {\mathfrak {g}}={\mathfrak {s}}\oplus {\mathfrak {a}};} there are alternative characterizations, given below. | wikipedia |
semi-simple lie group | In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra is a finite-dimensional Lie algebra over a field of characteristic 0. For such a Lie algebra g {\displaystyle {\mathfrak {g}}} , if nonzero, the following conditions are equivalent: g {\displaystyle {\mathfrak {g}}} is semisimple; the Killing form, κ(x,y) = tr(ad(x)ad(y)), is non-degenerate; g {\displaystyle {\mathfrak {g}}} has no non-zero abelian ideals; g {\displaystyle {\mathfrak {g}}} has no non-zero solvable ideals; the radical (maximal solvable ideal) of g {\displaystyle {\mathfrak {g}}} is zero. | wikipedia |
lie bialgebra | In mathematics, a Lie bialgebra is the Lie-theoretic case of a bialgebra: it is a set with a Lie algebra and a Lie coalgebra structure which are compatible. It is a bialgebra where the multiplication is skew-symmetric and satisfies a dual Jacobi identity, so that the dual vector space is a Lie algebra, whereas the comultiplication is a 1-cocycle, so that the multiplication and comultiplication are compatible. The cocycle condition implies that, in practice, one studies only classes of bialgebras that are cohomologous to a Lie bialgebra on a coboundary. They are also called Poisson-Hopf algebras, and are the Lie algebra of a Poisson–Lie group. Lie bialgebras occur naturally in the study of the Yang–Baxter equations. | wikipedia |
lie superalgebra | In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2‑grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. In most of these theories, the even elements of the superalgebra correspond to bosons and odd elements to fermions (but this is not always true; for example, the BRST supersymmetry is the other way around). | wikipedia |
lie-* algebra | In mathematics, a Lie-* algebra is a D-module with a Lie* bracket. They were introduced by Alexander Beilinson and Vladimir Drinfeld (Beilinson & Drinfeld (2004, section 2.5.3)), and are similar to the conformal algebras discussed by Kac (1998) and to vertex Lie algebras. | wikipedia |
maharam algebra | In mathematics, a Maharam algebra is a complete Boolean algebra with a continuous submeasure (defined below). They were introduced by Dorothy Maharam (1947). | wikipedia |
malcev lie algebra | In mathematics, a Malcev Lie algebra, or Mal'tsev Lie algebra, is a generalization of a rational nilpotent Lie algebra, and Malcev groups are similar. Both were introduced by Quillen (1969, Appendix A3), based on the work of (Mal'cev 1949). | wikipedia |
malcev algebra | In mathematics, a Malcev algebra (or Maltsev algebra or Moufang–Lie algebra) over a field is a nonassociative algebra that is antisymmetric, so that x y = − y x {\displaystyle xy=-yx} and satisfies the Malcev identity ( x y ) ( x z ) = ( ( x y ) z ) x + ( ( y z ) x ) x + ( ( z x ) x ) y . {\displaystyle (xy)(xz)=((xy)z)x+((yz)x)x+((zx)x)y.} They were first defined by Anatoly Maltsev (1955). | wikipedia |
malcev algebra | Malcev algebras play a role in the theory of Moufang loops that generalizes the role of Lie algebras in the theory of groups. Namely, just as the tangent space of the identity element of a Lie group forms a Lie algebra, the tangent space of the identity of a smooth Moufang loop forms a Malcev algebra. Moreover, just as a Lie group can be recovered from its Lie algebra under certain supplementary conditions, a smooth Moufang loop can be recovered from its Malcev algebra if certain supplementary conditions hold. For example, this is true for a connected, simply connected real-analytic Moufang loop. | wikipedia |
algebraic hyperbolicity | In mathematics, a Mordellic variety is an algebraic variety which has only finitely many points in any finitely generated field. The terminology was introduced by Serge Lang to enunciate a range of conjectures linking the geometry of varieties to their Diophantine properties. | wikipedia |
list of finite-dimensional nichols algebras | In mathematics, a Nichols algebra is a Hopf algebra in a braided category assigned to an object V in this category (e.g. a braided vector space). The Nichols algebra is a quotient of the tensor algebra of V enjoying a certain universal property and is typically infinite-dimensional. Nichols algebras appear naturally in any pointed Hopf algebra and enabled their classification in important cases. The most well known examples for Nichols algebras are the Borel parts U q ( g ) + {\displaystyle U_{q}({\mathfrak {g}})^{+}} of the infinite-dimensional quantum groups when q is no root of unity, and the first examples of finite-dimensional Nichols algebras are the Borel parts u q ( g ) + {\displaystyle u_{q}({\mathfrak {g}})^{+}} of the Frobenius–Lusztig kernel (small quantum group) when q is a root of unity. | wikipedia |
list of finite-dimensional nichols algebras | The following article lists all known finite-dimensional Nichols algebras B ( V ) {\displaystyle {\mathfrak {B}}(V)} where V {\displaystyle V} is a Yetter–Drinfel'd module over a finite group G {\displaystyle G} , where the group is generated by the support of V {\displaystyle V} . For more details on Nichols algebras see Nichols algebra. There are two major cases: G {\displaystyle G} abelian, which implies V {\displaystyle V} is diagonally braided x i ⊗ x j ↦ q i j x j ⊗ x i {\displaystyle x_{i}\otimes x_{j}\mapsto q_{ij}x_{j}\otimes x_{i}} . | wikipedia |
list of finite-dimensional nichols algebras | G {\displaystyle G} nonabelian. The rank is the number of irreducible summands V = ⨁ i ∈ I V i {\displaystyle V=\bigoplus _{i\in I}V_{i}} in the semisimple Yetter–Drinfel'd module V {\displaystyle V} . The irreducible summands V i = O χ {\displaystyle V_{i}={\mathcal {O}}_{}^{\chi }} are each associated to a conjugacy class ⊂ G {\displaystyle \subset G} and an irreducible representation χ {\displaystyle \chi } of the centralizer Cent ( g ) {\displaystyle \operatorname {Cent} (g)} . | wikipedia |
list of finite-dimensional nichols algebras | To any Nichols algebra there is by attached a generalized root system and a Weyl groupoid. These are classified in. | wikipedia |
list of finite-dimensional nichols algebras | In particular several Dynkin diagrams (for inequivalent types of Weyl chambers). Each Dynkin diagram has one vertex per irreducible V i {\displaystyle V_{i}} and edges depending on their braided commutators in the Nichols algebra. The Hilbert series of the graded algebra B ( V ) {\displaystyle {\mathfrak {B}}(V)} is given. | wikipedia |
list of finite-dimensional nichols algebras | An observation is that it factorizes in each case into polynomials ( n ) t := 1 + t + t 2 + ⋯ + t n − 1 {\displaystyle (n)_{t}:=1+t+t^{2}+\cdots +t^{n-1}} . We only give the Hilbert series and dimension of the Nichols algebra in characteristic 0 {\displaystyle 0} .Note that a Nichols algebra only depends on the braided vector space V {\displaystyle V} and can therefore be realized over many different groups. Sometimes there are two or three Nichols algebras with different V {\displaystyle V} and non-isomorphic Nichols algebra, which are closely related (e.g. cocycle twists of each other). These are given by different conjugacy classes in the same column. | wikipedia |
petersson algebra | In mathematics, a Petersson algebra is a composition algebra over a field constructed from an order-3 automorphism of a Hurwitz algebra. They were first constructed by Petersson (1969). | wikipedia |
poisson algebra | In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz's law; that is, the bracket is also a derivation. Poisson algebras appear naturally in Hamiltonian mechanics, and are also central in the study of quantum groups. Manifolds with a Poisson algebra structure are known as Poisson manifolds, of which the symplectic manifolds and the Poisson–Lie groups are a special case. The algebra is named in honour of Siméon Denis Poisson. | wikipedia |
poisson superbracket | In mathematics, a Poisson superalgebra is a Z2-graded generalization of a Poisson algebra. Specifically, a Poisson superalgebra is an (associative) superalgebra A with a Lie superbracket : A ⊗ A → A {\displaystyle :A\otimes A\to A} such that (A, ) is a Lie superalgebra and the operator : A → A {\displaystyle :A\to A} is a superderivation of A: = z + ( − 1 ) | x | | y | y . {\displaystyle =z+(-1)^{|x||y|}y.\,} A supercommutative Poisson algebra is one for which the (associative) product is supercommutative. This is one possible way of "super"izing the Poisson algebra. | wikipedia |
poisson superbracket | This gives the classical dynamics of fermion fields and classical spin-1/2 particles. The other is to define an antibracket algebra instead. This is used in the BRST and Batalin-Vilkovisky formalism. | wikipedia |
relevance vector machine | In mathematics, a Relevance Vector Machine (RVM) is a machine learning technique that uses Bayesian inference to obtain parsimonious solutions for regression and probabilistic classification. The RVM has an identical functional form to the support vector machine, but provides probabilistic classification. It is actually equivalent to a Gaussian process model with covariance function: k ( x , x ′ ) = ∑ j = 1 N 1 α j φ ( x , x j ) φ ( x ′ , x j ) {\displaystyle k(\mathbf {x} ,\mathbf {x'} )=\sum _{j=1}^{N}{\frac {1}{\alpha _{j}}}\varphi (\mathbf {x} ,\mathbf {x} _{j})\varphi (\mathbf {x} ',\mathbf {x} _{j})} where φ {\displaystyle \varphi } is the kernel function (usually Gaussian), α j {\displaystyle \alpha _{j}} are the variances of the prior on the weight vector w ∼ N ( 0 , α − 1 I ) {\displaystyle w\sim N(0,\alpha ^{-1}I)} , and x 1 , … , x N {\displaystyle \mathbf {x} _{1},\ldots ,\mathbf {x} _{N}} are the input vectors of the training set.Compared to that of support vector machines (SVM), the Bayesian formulation of the RVM avoids the set of free parameters of the SVM (that usually require cross-validation-based post-optimizations). However RVMs use an expectation maximization (EM)-like learning method and are therefore at risk of local minima. This is unlike the standard sequential minimal optimization (SMO)-based algorithms employed by SVMs, which are guaranteed to find a global optimum (of the convex problem). The relevance vector machine was patented in the United States by Microsoft (patent expired September 4, 2019). | wikipedia |
ringel–hall algebra | In mathematics, a Ringel–Hall algebra is a generalization of the Hall algebra, studied by Claus Michael Ringel (1990). It has a basis of equivalence classes of objects of an abelian category, and the structure constants for this basis are related to the numbers of extensions of objects in the category. | wikipedia |
rota–baxter algebra | In mathematics, a Rota–Baxter algebra is an associative algebra, together with a particular linear map R which satisfies the Rota–Baxter identity. It appeared first in the work of the American mathematician Glen E. Baxter in the realm of probability theory. Baxter's work was further explored from different angles by Gian-Carlo Rota, Pierre Cartier, and Frederic V. Atkinson, among others. Baxter’s derivation of this identity that later bore his name emanated from some of the fundamental results of the famous probabilist Frank Spitzer in random walk theory.In the 1980s, the Rota-Baxter operator of weight 0 in the context of Lie algebras was rediscovered as the operator form of the classical Yang–Baxter equation, named after the well-known physicists Chen-Ning Yang and Rodney Baxter. The study of Rota–Baxter algebras experienced a renaissance this century, beginning with several developments, in the algebraic approach to renormalization of perturbative quantum field theory, dendriform algebras, associative analogue of the classical Yang–Baxter equation and mixable shuffle product constructions. | wikipedia |
schur-concave function | In mathematics, a Schur-convex function, also known as S-convex, isotonic function and order-preserving function is a function f: R d → R {\displaystyle f:\mathbb {R} ^{d}\rightarrow \mathbb {R} } that for all x , y ∈ R d {\displaystyle x,y\in \mathbb {R} ^{d}} such that x {\displaystyle x} is majorized by y {\displaystyle y} , one has that f ( x ) ≤ f ( y ) {\displaystyle f(x)\leq f(y)} . Named after Issai Schur, Schur-convex functions are used in the study of majorization. Every function that is convex and symmetric is also Schur-convex. The opposite implication is not true, but all Schur-convex functions are symmetric (under permutations of the arguments). | wikipedia |
schwartz–bruhat function | In mathematics, a Schwartz–Bruhat function, named after Laurent Schwartz and François Bruhat, is a complex valued function on a locally compact abelian group, such as the adeles, that generalizes a Schwartz function on a real vector space. A tempered distribution is defined as a continuous linear functional on the space of Schwartz–Bruhat functions. | wikipedia |
shintani zeta function | In mathematics, a Shintani zeta function or Shintani L-function is a generalization of the Riemann zeta function. They were first studied by Takuro Shintani (1976). They include Hurwitz zeta functions and Barnes zeta functions. | wikipedia |
stone algebra | In mathematics, a Stone algebra, or Stone lattice, is a pseudo-complemented distributive lattice such that a* ∨ a** = 1. They were introduced by Grätzer & Schmidt (1957) and named after Marshall Harvey Stone. Boolean algebras are Stone algebras, and Stone algebras are Ockham algebras. Examples: The open-set lattice of an extremally disconnected space is a Stone algebra. The lattice of positive divisors of a given positive integer is a Stone lattice. | wikipedia |
suslin algebra | In mathematics, a Suslin algebra is a Boolean algebra that is complete, atomless, countably distributive, and satisfies the countable chain condition. They are named after Mikhail Yakovlevich Suslin.The existence of Suslin algebras is independent of the axioms of ZFC, and is equivalent to the existence of Suslin trees or Suslin lines. | wikipedia |
swiss cheese (mathematics) | In mathematics, a Swiss cheese is a compact subset of the complex plane obtained by removing from a closed disc some countable union of open discs, usually with some restriction on the centres and radii of the removed discs. Traditionally the deleted discs should have pairwise disjoint closures which are subsets of the interior of the starting disc, the sum of the radii of the deleted discs should be finite, and the Swiss cheese should have empty interior. This is the type of Swiss cheese originally introduced by the Swiss mathematician Alice Roth. More generally, a Swiss cheese may be all or part of Euclidean space Rn – or of an even more complicated manifold – with "holes" in it. | wikipedia |
takiff algebra | In mathematics, a Takiff algebra is a Lie algebra over a truncated polynomial ring. More precisely, a Takiff algebra of a Lie algebra g over a field k is a Lie algebra of the form g/(xn+1) = g⊗kk/(xn+1) for some positive integer n. Sometimes these are called generalized Takiff algebras, and the name Takiff algebra is used for the case when n = 1. These algebras (for n = 1) were studied by Takiff (1971), who in some cases described the ring of polynomials on these algebras invariant under the action of the adjoint group. | wikipedia |
verlinde formula | In mathematics, a Verlinde algebra is a finite-dimensional associative algebra introduced by Erik Verlinde (1988), with a basis of elements φλ corresponding to primary fields of a rational two-dimensional conformal field theory, whose structure constants Nνλμ describe fusion of primary fields. | wikipedia |
whittaker function | In mathematics, a Whittaker function is a special solution of Whittaker's equation, a modified form of the confluent hypergeometric equation introduced by Whittaker (1903) to make the formulas involving the solutions more symmetric. More generally, Jacquet (1966, 1967) introduced Whittaker functions of reductive groups over local fields, where the functions studied by Whittaker are essentially the case where the local field is the real numbers and the group is SL2(R). Whittaker's equation is d 2 w d z 2 + ( − 1 4 + κ z + 1 / 4 − μ 2 z 2 ) w = 0. | wikipedia |
whittaker function | {\displaystyle {\frac {d^{2}w}{dz^{2}}}+\left(-{\frac {1}{4}}+{\frac {\kappa }{z}}+{\frac {1/4-\mu ^{2}}{z^{2}}}\right)w=0.} It has a regular singular point at 0 and an irregular singular point at ∞. Two solutions are given by the Whittaker functions Mκ,μ(z), Wκ,μ(z), defined in terms of Kummer's confluent hypergeometric functions M and U by M κ , μ ( z ) = exp ( − z / 2 ) z μ + 1 2 M ( μ − κ + 1 2 , 1 + 2 μ , z ) {\displaystyle M_{\kappa ,\mu }\left(z\right)=\exp \left(-z/2\right)z^{\mu +{\tfrac {1}{2}}}M\left(\mu -\kappa +{\tfrac {1}{2}},1+2\mu ,z\right)} W κ , μ ( z ) = exp ( − z / 2 ) z μ + 1 2 U ( μ − κ + 1 2 , 1 + 2 μ , z ) . | wikipedia |
whittaker function | {\displaystyle W_{\kappa ,\mu }\left(z\right)=\exp \left(-z/2\right)z^{\mu +{\tfrac {1}{2}}}U\left(\mu -\kappa +{\tfrac {1}{2}},1+2\mu ,z\right).} The Whittaker function W κ , μ ( z ) {\displaystyle W_{\kappa ,\mu }(z)} is the same as those with opposite values of μ, in other words considered as a function of μ at fixed κ and z it is even functions. When κ and z are real, the functions give real values for real and imaginary values of μ. These functions of μ play a role in so-called Kummer spaces.Whittaker functions appear as coefficients of certain representations of the group SL2(R), called Whittaker models. | wikipedia |
zariski geometry | In mathematics, a Zariski geometry consists of an abstract structure introduced by Ehud Hrushovski and Boris Zilber, in order to give a characterisation of the Zariski topology on an algebraic curve, and all its powers. The Zariski topology on a product of algebraic varieties is very rarely the product topology, but richer in closed sets defined by equations that mix two sets of variables. The result described gives that a very definite meaning, applying to projective curves and compact Riemann surfaces in particular. | wikipedia |
zinbiel algebra | In mathematics, a Zinbiel algebra or dual Leibniz algebra is a module over a commutative ring with a bilinear product satisfying the defining identity: ( a ∘ b ) ∘ c = a ∘ ( b ∘ c ) + a ∘ ( c ∘ b ) . {\displaystyle (a\circ b)\circ c=a\circ (b\circ c)+a\circ (c\circ b).} Zinbiel algebras were introduced by Jean-Louis Loday (1995). | wikipedia |
zinbiel algebra | The name was proposed by Jean-Michel Lemaire as being "opposite" to Leibniz algebra.In any Zinbiel algebra, the symmetrised product a ⋆ b = a ∘ b + b ∘ a {\displaystyle a\star b=a\circ b+b\circ a} is associative. A Zinbiel algebra is the Koszul dual concept to a Leibniz algebra. The free Zinbiel algebra over V is the tensor algebra with product ( x 0 ⊗ ⋯ ⊗ x p ) ∘ ( x p + 1 ⊗ ⋯ ⊗ x p + q ) = x 0 ∑ ( p , q ) ( x 1 , … , x p + q ) , {\displaystyle (x_{0}\otimes \cdots \otimes x_{p})\circ (x_{p+1}\otimes \cdots \otimes x_{p+q})=x_{0}\sum _{(p,q)}(x_{1},\ldots ,x_{p+q}),} where the sum is over all ( p , q ) {\displaystyle (p,q)} shuffles. | wikipedia |
4-ball (mathematics) | In mathematics, a ball is the solid figure bounded by a sphere; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defined not only in three-dimensional Euclidean space but also for lower and higher dimensions, and for metric spaces in general. A ball in n dimensions is called a hyperball or n-ball and is bounded by a hypersphere or (n−1)-sphere. | wikipedia |
4-ball (mathematics) | Thus, for example, a ball in the Euclidean plane is the same thing as a disk, the area bounded by a circle. In Euclidean 3-space, a ball is taken to be the volume bounded by a 2-dimensional sphere. | wikipedia |
4-ball (mathematics) | In a one-dimensional space, a ball is a line segment. In other contexts, such as in Euclidean geometry and informal use, sphere is sometimes used to mean ball. In the field of topology the closed n {\displaystyle n} -dimensional ball is often denoted as B n {\displaystyle B^{n}} or D n {\displaystyle D^{n}} while the open n {\displaystyle n} -dimensional ball is Int B n {\displaystyle \operatorname {Int} B^{n}} or Int D n {\displaystyle \operatorname {Int} D^{n}} . | wikipedia |
band (algebra) | In mathematics, a band (also called idempotent semigroup) is a semigroup in which every element is idempotent (in other words equal to its own square). Bands were first studied and named by A. H. Clifford (1954). The lattice of varieties of bands was described independently in the early 1970s by Biryukov, Fennemore and Gerhard. Semilattices, left-zero bands, right-zero bands, rectangular bands, normal bands, left-regular bands, right-regular bands and regular bands are specific subclasses of bands that lie near the bottom of this lattice and which are of particular interest; they are briefly described below. | wikipedia |
countable base | In mathematics, a base (or basis; PL: bases) for the topology τ of a topological space (X, τ) is a family B {\displaystyle {\mathcal {B}}} of open subsets of X such that every open set of the topology is equal to the union of some sub-family of B {\displaystyle {\mathcal {B}}} . For example, the set of all open intervals in the real number line R {\displaystyle \mathbb {R} } is a basis for the Euclidean topology on R {\displaystyle \mathbb {R} } because every open interval is an open set, and also every open subset of R {\displaystyle \mathbb {R} } can be written as a union of some family of open intervals. Bases are ubiquitous throughout topology. The sets in a base for a topology, which are called basic open sets, are often easier to describe and use than arbitrary open sets. | wikipedia |
countable base | Many important topological definitions such as continuity and convergence can be checked using only basic open sets instead of arbitrary open sets. Some topologies have a base of open sets with specific useful properties that may make checking such topological definitions easier. Not all families of subsets of a set X {\displaystyle X} form a base for a topology on X {\displaystyle X} . | wikipedia |
countable base | Under some conditions detailed below, a family of subsets will form a base for a (unique) topology on X {\displaystyle X} , obtained by taking all possible unions of subfamilies. Such families of sets are very frequently used to define topologies. A weaker notion related to bases is that of a subbase for a topology. Bases for topologies are also closely related to neighborhood bases. | wikipedia |
algebraic operation | In mathematics, a basic algebraic operation is any one of the common operations of arithmetic, which include addition, subtraction, multiplication, division, raising to a whole number power, and taking roots (fractional power). These operations may be performed on numbers, in which case they are often called arithmetic operations. They may also be performed, in a similar way, on variables, algebraic expressions, and more generally, on elements of algebraic structures, such as groups and fields. | wikipedia |
algebraic operation | An algebraic operation may also be defined simply as a function from a Cartesian power of a set to the same set.The term algebraic operation may also be used for operations that may be defined by compounding basic algebraic operations, such as the dot product. In calculus and mathematical analysis, algebraic operation is also used for the operations that may be defined by purely algebraic methods. For example, exponentiation with an integer or rational exponent is an algebraic operation, but not the general exponentiation with a real or complex exponent. Also, the derivative is an operation that is not algebraic. | wikipedia |
semi-algebraic sets | In mathematics, a basic semialgebraic set is a set defined by polynomial equalities and polynomial inequalities, and a semialgebraic set is a finite union of basic semialgebraic sets. A semialgebraic function is a function with a semialgebraic graph. Such sets and functions are mainly studied in real algebraic geometry which is the appropriate framework for algebraic geometry over the real numbers. | wikipedia |
blending function | In mathematics, a basis function is an element of a particular basis for a function space. Every function in the function space can be represented as a linear combination of basis functions, just as every vector in a vector space can be represented as a linear combination of basis vectors. In numerical analysis and approximation theory, basis functions are also called blending functions, because of their use in interpolation: In this application, a mixture of the basis functions provides an interpolating function (with the "blend" depending on the evaluation of the basis functions at the data points). | wikipedia |
bialgebra | In mathematics, a bialgebra over a field K is a vector space over K which is both a unital associative algebra and a counital coassociative coalgebra. The algebraic and coalgebraic structures are made compatible with a few more axioms. Specifically, the comultiplication and the counit are both unital algebra homomorphisms, or equivalently, the multiplication and the unit of the algebra both are coalgebra morphisms. (These statements are equivalent since they are expressed by the same commutative diagrams.) | wikipedia |
bialgebra | Similar bialgebras are related by bialgebra homomorphisms. A bialgebra homomorphism is a linear map that is both an algebra and a coalgebra homomorphism. As reflected in the symmetry of the commutative diagrams, the definition of bialgebra is self-dual, so if one can define a dual of B (which is always possible if B is finite-dimensional), then it is automatically a bialgebra. | wikipedia |
binary functions | In mathematics, a binary function (also called bivariate function, or function of two variables) is a function that takes two inputs. Precisely stated, a function f {\displaystyle f} is binary if there exists sets X , Y , Z {\displaystyle X,Y,Z} such that f: X × Y → Z {\displaystyle \,f\colon X\times Y\rightarrow Z} where X × Y {\displaystyle X\times Y} is the Cartesian product of X {\displaystyle X} and Y . {\displaystyle Y.} | wikipedia |
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), 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. | wikipedia |
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. | wikipedia |
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) ∈ (<)". | wikipedia |
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. | wikipedia |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.