text
stringlengths 9
3.55k
| source
stringlengths 31
280
|
|---|---|
In mathematics, a Hankel contour is a path in the complex plane which extends from (+∞,δ), around the origin counter clockwise and back to (+∞,−δ), where δ is an arbitrarily small positive number. The contour thus remains arbitrarily close to the real axis but without crossing the real axis except for negative values of x. The Hankel contour can also be represented by a path that has mirror images just above and below the real axis, connected to a circle of radius ε, centered at the origin, where ε is an arbitrarily small number. The two linear portions of the contour are said to be a distance of δ from the real axis. Thus, the total distance between the linear portions of the contour is 2δ.
|
https://en.wikipedia.org/wiki/Hankel_contour
|
The contour is traversed in the positively-oriented sense, meaning that the circle around the origin is traversed counter-clockwise. Use of Hankel contours is one of the methods of contour integration. This type of path for contour integrals was first used by Hermann Hankel in his investigations of the Gamma function. The Hankel contour is used to evaluate integrals such as the Gamma function, the Riemann zeta function, and other Hankel functions (which are Bessel functions of the third kind).
|
https://en.wikipedia.org/wiki/Hankel_contour
|
In mathematics, a Hardy field is a field consisting of germs of real-valued functions at infinity that are closed under differentiation. They are named after the English mathematician G. H. Hardy.
|
https://en.wikipedia.org/wiki/Hardy_field
|
In mathematics, a Hartshorne ellipse is an ellipse in the unit ball bounded by the 4-sphere S4 such that the ellipse and the circle given by intersection of its plane with S4 satisfy the Poncelet condition that there is a triangle with vertices on the circle and edges tangent to the ellipse. They were introduced by Hartshorne (1978), who showed that they correspond to k = 2 instantons on S4.
|
https://en.wikipedia.org/wiki/Hartshorne_ellipse
|
In mathematics, a Hasse–Schmidt derivation is an extension of the notion of a derivation. The concept was introduced by Schmidt & Hasse (1937).
|
https://en.wikipedia.org/wiki/Hasse–Schmidt_derivation
|
In mathematics, a Hausdorff gap consists roughly of two collections of sequences of integers, such that there is no sequence lying between the two collections. The first example was found by Hausdorff (1909). The existence of Hausdorff gaps shows that the partially ordered set of possible growth rates of sequences is not complete.
|
https://en.wikipedia.org/wiki/Hausdorff_gap
|
In mathematics, a Hausdorff space X is called a fixed-point space if every continuous function f: X → X {\displaystyle f:X\rightarrow X} has a fixed point. For example, any closed interval in R {\displaystyle \mathbb {R} } is a fixed point space, and it can be proved from the intermediate value property of real continuous function. The open interval (a, b), however, is not a fixed point space.
|
https://en.wikipedia.org/wiki/Fixed-point_space
|
To see it, consider the function f ( x ) = a + 1 b − a ⋅ ( x − a ) 2 {\displaystyle f(x)=a+{\frac {1}{b-a}}\cdot (x-a)^{2}} , for example. Any linearly ordered space that is connected and has a top and a bottom element is a fixed point space. Note that, in the definition, we could easily have disposed of the condition that the space is Hausdorff.
|
https://en.wikipedia.org/wiki/Fixed-point_space
|
In mathematics, a Hausdorff space is said to be H-closed, or Hausdorff closed, or absolutely closed if it is closed in every Hausdorff space containing it as a subspace. This property is a generalization of compactness, since a compact subset of a Hausdorff space is closed. Thus, every compact Hausdorff space is H-closed. The notion of an H-closed space has been introduced in 1924 by P. Alexandroff and P. Urysohn.
|
https://en.wikipedia.org/wiki/H-closed_space
|
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
|
https://en.wikipedia.org/wiki/Hecke_algebra_(disambiguation)
|
In mathematics, a Hecke algebra of a locally compact group is an algebra of bi-invariant measures under convolution.
|
https://en.wikipedia.org/wiki/Hecke_algebra_of_a_locally_compact_group
|
In mathematics, a Heegner point is a point on a modular curve that is the image of a quadratic imaginary point of the upper half-plane. They were defined by Bryan Birch and named after Kurt Heegner, who used similar ideas to prove Gauss's conjecture on imaginary quadratic fields of class number one.
|
https://en.wikipedia.org/wiki/Heegner_point
|
In mathematics, a Heilbronn set is an infinite set S of natural numbers for which every real number can be arbitrarily closely approximated by a fraction whose denominator is in S. For any given real number θ {\displaystyle \theta } and natural number h {\displaystyle h} , it is easy to find the integer g {\displaystyle g} such that g / h {\displaystyle g/h} is closest to θ {\displaystyle \theta } . For example, for the real number π {\displaystyle \pi } and h = 100 {\displaystyle h=100} we have g = 314 {\displaystyle g=314} . If we call the closeness of θ {\displaystyle \theta } to g / h {\displaystyle g/h} the difference between h θ {\displaystyle h\theta } and g {\displaystyle g} , the closeness is always less than 1/2 (in our example it is 0.15926...). A collection of numbers is a Heilbronn set if for any θ {\displaystyle \theta } we can always find a sequence of values for h {\displaystyle h} in the set where the closeness tends to zero. More mathematically let ‖ α ‖ {\displaystyle \|\alpha \|} denote the distance from α {\displaystyle \alpha } to the nearest integer then H {\displaystyle {\mathcal {H}}} is a Heilbronn set if and only if for every real number θ {\displaystyle \theta } and every ε > 0 {\displaystyle \varepsilon >0} there exists h ∈ H {\displaystyle h\in {\mathcal {H}}} such that ‖ h θ ‖ < ε {\displaystyle \|h\theta \|<\varepsilon } .
|
https://en.wikipedia.org/wiki/Heilbronn_set
|
In mathematics, a Henselian ring (or Hensel ring) is a local ring in which Hensel's lemma holds. They were introduced by Azumaya (1951), who named them after Kurt Hensel. Azumaya originally allowed Henselian rings to be non-commutative, but most authors now restrict them to be commutative. Some standard references for Hensel rings are (Nagata 1975, Chapter VII), (Raynaud 1970), and (Grothendieck 1967, Chapter 18).
|
https://en.wikipedia.org/wiki/Henselian_field
|
In mathematics, a Hermitian connection ∇ {\displaystyle \nabla } is a connection on a Hermitian vector bundle E {\displaystyle E} over a smooth manifold M {\displaystyle M} which is compatible with the Hermitian metric ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } on E {\displaystyle E} , meaning that v ⟨ s , t ⟩ = ⟨ ∇ v s , t ⟩ + ⟨ s , ∇ v t ⟩ {\displaystyle v\langle s,t\rangle =\langle \nabla _{v}s,t\rangle +\langle s,\nabla _{v}t\rangle } for all smooth vector fields v {\displaystyle v} and all smooth sections s , t {\displaystyle s,t} of E {\displaystyle E} . If X {\displaystyle X} is a complex manifold, and the Hermitian vector bundle E {\displaystyle E} on X {\displaystyle X} is equipped with a holomorphic structure, then there is a unique Hermitian connection whose (0, 1)-part coincides with the Dolbeault operator ∂ ¯ E {\displaystyle {\bar {\partial }}_{E}} on E {\displaystyle E} associated to the holomorphic structure. This is called the Chern connection on E {\displaystyle E} . The curvature of the Chern connection is a (1, 1)-form. For details, see Hermitian metrics on a holomorphic vector bundle. In particular, if the base manifold is Kähler and the vector bundle is its tangent bundle, then the Chern connection coincides with the Levi-Civita connection of the associated Riemannian metric.
|
https://en.wikipedia.org/wiki/Hermitian_connection
|
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column, for all indices i and j: or in matrix form: Hermitian matrices can be understood as the complex extension of real symmetric matrices. If the conjugate transpose of a matrix A {\displaystyle A} is denoted by A H , {\displaystyle A^{\mathsf {H}},} then the Hermitian property can be written concisely as Hermitian matrices are named after Charles Hermite, who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of always having real eigenvalues. Other, equivalent notations in common use are A H = A † = A ∗ , {\displaystyle A^{\mathsf {H}}=A^{\dagger }=A^{\ast },} although in quantum mechanics, A ∗ {\displaystyle A^{\ast }} typically means the complex conjugate only, and not the conjugate transpose.
|
https://en.wikipedia.org/wiki/Hermitian_matrices
|
In mathematics, a Hermitian symmetric space is a Hermitian manifold which at every point has an inversion symmetry preserving the Hermitian structure. First studied by Élie Cartan, they form a natural generalization of the notion of Riemannian symmetric space from real manifolds to complex manifolds. Every Hermitian symmetric space is a homogeneous space for its isometry group and has a unique decomposition as a product of irreducible spaces and a Euclidean space.
|
https://en.wikipedia.org/wiki/Hermitian_symmetric_domain
|
The irreducible spaces arise in pairs as a non-compact space that, as Borel showed, can be embedded as an open subspace of its compact dual space. Harish Chandra showed that each non-compact space can be realized as a bounded symmetric domain in a complex vector space. The simplest case involves the groups SU(2), SU(1,1) and their common complexification SL(2,C).
|
https://en.wikipedia.org/wiki/Hermitian_symmetric_domain
|
In this case the non-compact space is the unit disk, a homogeneous space for SU(1,1). It is a bounded domain in the complex plane C. The one-point compactification of C, the Riemann sphere, is the dual space, a homogeneous space for SU(2) and SL(2,C). Irreducible compact Hermitian symmetric spaces are exactly the homogeneous spaces of simple compact Lie groups by maximal closed connected subgroups which contain a maximal torus and have center isomorphic to the circle group. There is a complete classification of irreducible spaces, with four classical series, studied by Cartan, and two exceptional cases; the classification can be deduced from Borel–de Siebenthal theory, which classifies closed connected subgroups containing a maximal torus. Hermitian symmetric spaces appear in the theory of Jordan triple systems, several complex variables, complex geometry, automorphic forms and group representations, in particular permitting the construction of the holomorphic discrete series representations of semisimple Lie groups.
|
https://en.wikipedia.org/wiki/Hermitian_symmetric_domain
|
In mathematics, a Hessian pair or Hessian duad, named for Otto Hesse, is a pair of points of the projective line canonically associated with a set of 3 points of the projective line. More generally, one can define the Hessian pair of any triple of elements from a set that can be identified with a projective line, such as a rational curve, a pencil of divisors, a pencil of lines, and so on.
|
https://en.wikipedia.org/wiki/Hessian_pair
|
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.
|
https://en.wikipedia.org/wiki/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.
|
https://en.wikipedia.org/wiki/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.
|
https://en.wikipedia.org/wiki/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.
|
https://en.wikipedia.org/wiki/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.
|
https://en.wikipedia.org/wiki/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.
|
https://en.wikipedia.org/wiki/Heyting_algebra
|
In mathematics, a Higgs bundle is a pair ( E , φ ) {\displaystyle (E,\varphi )} consisting of a holomorphic vector bundle E and a Higgs field φ {\displaystyle \varphi } , a holomorphic 1-form taking values in the bundle of endomorphisms of E such that φ ∧ φ = 0 {\displaystyle \varphi \wedge \varphi =0} . Such pairs were introduced by Nigel Hitchin (1987), who named the field φ {\displaystyle \varphi } after Peter Higgs because of an analogy with Higgs bosons. The term 'Higgs bundle', and the condition φ ∧ φ = 0 {\displaystyle \varphi \wedge \varphi =0} (which is vacuous in Hitchin's original set-up on Riemann surfaces) was introduced later by Carlos Simpson.A Higgs bundle can be thought of as a "simplified version" of a flat holomorphic connection on a holomorphic vector bundle, where the derivative is scaled to zero. The nonabelian Hodge correspondence says that, under suitable stability conditions, the category of flat holomorphic connections on a smooth, projective complex algebraic variety, the category of representations of the fundamental group of the variety, and the category of Higgs bundles over this variety are actually equivalent. Therefore, one can deduce results about gauge theory with flat connections by working with the simpler Higgs bundles.
|
https://en.wikipedia.org/wiki/Higgs_bundle
|
In mathematics, a Hilbert manifold is a manifold modeled on Hilbert spaces. Thus it is a separable Hausdorff space in which each point has a neighbourhood homeomorphic to an infinite dimensional Hilbert space. The concept of a Hilbert manifold provides a possibility of extending the theory of manifolds to infinite-dimensional setting. Analogously to the finite-dimensional situation, one can define a differentiable Hilbert manifold by considering a maximal atlas in which the transition maps are differentiable.
|
https://en.wikipedia.org/wiki/Hilbert_bundle
|
In mathematics, a Hilbert modular form is a generalization of modular forms to functions of two or more variables. It is a (complex) analytic function on the m-fold product of upper half-planes H {\displaystyle {\mathcal {H}}} satisfying a certain kind of functional equation.
|
https://en.wikipedia.org/wiki/Hilbert_modular_form
|
In mathematics, a Hilbert modular surface or Hilbert–Blumenthal surface is an algebraic surface obtained by taking a quotient of a product of two copies of the upper half-plane by a Hilbert modular group. More generally, a Hilbert modular variety is an algebraic variety obtained by taking a quotient of a product of multiple copies of the upper half-plane by a Hilbert modular group. Hilbert modular surfaces were first described by Otto Blumenthal (1903, 1904) using some unpublished notes written by David Hilbert about 10 years before.
|
https://en.wikipedia.org/wiki/Hilbert_modular_surface
|
In mathematics, a Hilbert–Schmidt integral operator is a type of integral transform. Specifically, given a domain (an open and connected set) Ω in n-dimensional Euclidean space Rn, a Hilbert–Schmidt kernel is a function k: Ω × Ω → C with ∫ Ω ∫ Ω | k ( x , y ) | 2 d x d y < ∞ {\displaystyle \int _{\Omega }\int _{\Omega }|k(x,y)|^{2}\,dx\,dy<\infty } (that is, the L2(Ω×Ω; C) norm of k is finite), and the associated Hilbert–Schmidt integral operator is the operator K: L2(Ω; C) → L2(Ω; C) given by ( K u ) ( x ) = ∫ Ω k ( x , y ) u ( y ) d y . {\displaystyle (Ku)(x)=\int _{\Omega }k(x,y)u(y)\,dy.} Then K is a Hilbert–Schmidt operator with Hilbert–Schmidt norm ‖ K ‖ H S = ‖ k ‖ L 2 .
|
https://en.wikipedia.org/wiki/Hilbert–Schmidt_integral_operator
|
{\displaystyle \Vert K\Vert _{\mathrm {HS} }=\Vert k\Vert _{L^{2}}.} Hilbert–Schmidt integral operators are both continuous (and hence bounded) and compact (as with all Hilbert–Schmidt operators). The concept of a Hilbert–Schmidt operator may be extended to any locally compact Hausdorff spaces.
|
https://en.wikipedia.org/wiki/Hilbert–Schmidt_integral_operator
|
Specifically, let X be a locally compact Hausdorff space equipped with a positive Borel measure. Suppose further that L2(X) is a separable Hilbert space. The above condition on the kernel k on Rn can be interpreted as demanding k belong to L2(X × X).
|
https://en.wikipedia.org/wiki/Hilbert–Schmidt_integral_operator
|
Then the operator ( K f ) ( x ) = ∫ X k ( x , y ) f ( y ) d y {\displaystyle (Kf)(x)=\int _{X}k(x,y)f(y)\,dy} is compact. If k ( x , y ) = k ( y , x ) ¯ {\displaystyle k(x,y)={\overline {k(y,x)}}} then K is also self-adjoint and so the spectral theorem applies. This is one of the fundamental constructions of such operators, which often reduces problems about infinite-dimensional vector spaces to questions about well-understood finite-dimensional eigenspaces. See Chapter 2 of the book by Bump in the references for examples.
|
https://en.wikipedia.org/wiki/Hilbert–Schmidt_integral_operator
|
In mathematics, a Hilbert–Schmidt operator, named after David Hilbert and Erhard Schmidt, is a bounded operator A: H → H {\displaystyle A\colon H\to H} that acts on a Hilbert space H {\displaystyle H} and has finite Hilbert–Schmidt norm where { e i: i ∈ I } {\displaystyle \{e_{i}:i\in I\}} is an orthonormal basis. The index set I {\displaystyle I} need not be countable. However, the sum on the right must contain at most countably many non-zero terms, to have meaning. This definition is independent of the choice of the orthonormal basis. In finite-dimensional Euclidean space, the Hilbert–Schmidt norm ‖ ⋅ ‖ HS {\displaystyle \|\cdot \|_{\text{HS}}} is identical to the Frobenius norm.
|
https://en.wikipedia.org/wiki/Hilbert–Schmidt_operator
|
In mathematics, a Hironaka decomposition is a representation of an algebra over a field as a finitely generated free module over a polynomial subalgebra or a regular local ring. Such decompositions are named after Heisuke Hironaka, who used this in his unpublished master's thesis at Kyoto University (Nagata 1962, p.217). Hironaka's criterion (Nagata 1962, theorem 25.16), sometimes called miracle flatness, states that a local ring R that is a finitely generated module over a regular Noetherian local ring S is Cohen–Macaulay if and only if it is a free module over S. There is a similar result for rings that are graded over a field rather than local.
|
https://en.wikipedia.org/wiki/Miracle_flatness
|
In mathematics, a Hirzebruch surface is a ruled surface over the projective line. They were studied by Friedrich Hirzebruch (1951).
|
https://en.wikipedia.org/wiki/Hirzebruch_surface
|
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.
|
https://en.wikipedia.org/wiki/Hodge_algebra
|
In mathematics, a Hodge structure, named after W. V. D. Hodge, is an algebraic structure at the level of linear algebra, similar to the one that Hodge theory gives to the cohomology groups of a smooth and compact Kähler manifold. Hodge structures have been generalized for all complex varieties (even if they are singular and non-complete) in the form of mixed Hodge structures, defined by Pierre Deligne (1970). A variation of Hodge structure is a family of Hodge structures parameterized by a manifold, first studied by Phillip Griffiths (1968). All these concepts were further generalized to mixed Hodge modules over complex varieties by Morihiko Saito (1989).
|
https://en.wikipedia.org/wiki/Hodge_filtration
|
In mathematics, a Hodge–Tate module is an analogue of a Hodge structure over p-adic fields. Serre (1967) introduced and named Hodge–Tate structures using the results of Tate (1967) on p-divisible groups.
|
https://en.wikipedia.org/wiki/Hodge-Tate_theory
|
In mathematics, a Hofstadter sequence is a member of a family of related integer sequences defined by non-linear recurrence relations.
|
https://en.wikipedia.org/wiki/Hofstadter_sequence
|
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)} .
|
https://en.wikipedia.org/wiki/Quasitriangular_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.
|
https://en.wikipedia.org/wiki/Hopf_algebra
|
In mathematics, a Hopfian group is a group G for which every epimorphism G → Gis an isomorphism. Equivalently, a group is Hopfian if and only if it is not isomorphic to any of its proper quotients. A group G is co-Hopfian if every monomorphism G → Gis an isomorphism. Equivalently, G is not isomorphic to any of its proper subgroups.
|
https://en.wikipedia.org/wiki/Hopfian_group
|
In mathematics, a Hughes plane is one of the non-Desarguesian projective planes found by Hughes (1957). There are examples of order p2n for every odd prime p and every positive integer n.
|
https://en.wikipedia.org/wiki/Hughes_plane
|
In mathematics, a Hurewicz space is a topological space that satisfies a certain basic selection principle that generalizes σ-compactness. A Hurewicz space is a space in which for every sequence of open covers U 1 , U 2 , … {\displaystyle {\mathcal {U}}_{1},{\mathcal {U}}_{2},\ldots } of the space there are finite sets F 1 ⊂ U 1 , F 2 ⊂ U 2 , … {\displaystyle {\mathcal {F}}_{1}\subset {\mathcal {U}}_{1},{\mathcal {F}}_{2}\subset {\mathcal {U}}_{2},\ldots } such that every point of the space belongs to all but finitely many sets ⋃ F 1 , ⋃ F 2 , … {\displaystyle \bigcup {\mathcal {F}}_{1},\bigcup {\mathcal {F}}_{2},\ldots } .
|
https://en.wikipedia.org/wiki/Hurewicz_space
|
In mathematics, a Hurwitz matrix, or Routh–Hurwitz matrix, in engineering stability matrix, is a structured real square matrix constructed with coefficients of a real polynomial.
|
https://en.wikipedia.org/wiki/Hurwitz_matrix
|
In mathematics, a Hurwitz polynomial, named after Adolf Hurwitz, is a polynomial whose roots (zeros) are located in the left half-plane of the complex plane or on the imaginary axis, that is, the real part of every root is zero or negative. Such a polynomial must have coefficients that are positive real numbers. The term is sometimes restricted to polynomials whose roots have real parts that are strictly negative, excluding the imaginary axis (i.e., a Hurwitz stable polynomial).A polynomial function P(s) of a complex variable s is said to be Hurwitz if the following conditions are satisfied: 1.
|
https://en.wikipedia.org/wiki/Hurwitz_polynomial
|
P(s) is real when s is real.2. The roots of P(s) have real parts which are zero or negative.Hurwitz polynomials are important in control systems theory, because they represent the characteristic equations of stable linear systems. Whether a polynomial is Hurwitz can be determined by solving the equation to find the roots, or from the coefficients without solving the equation by the Routh–Hurwitz stability criterion.
|
https://en.wikipedia.org/wiki/Hurwitz_polynomial
|
In mathematics, a Hurwitz quaternion (or Hurwitz integer) is a quaternion whose components are either all integers or all half-integers (halves of odd integers; a mixture of integers and half-integers is excluded). The set of all Hurwitz quaternions is H = { a + b i + c j + d k ∈ H ∣ a , b , c , d ∈ Z or a , b , c , d ∈ Z + 1 2 } . {\displaystyle H=\left\{a+bi+cj+dk\in \mathbb {H} \mid a,b,c,d\in \mathbb {Z} \;{\mbox{ or }}\,a,b,c,d\in \mathbb {Z} +{\tfrac {1}{2}}\right\}.}
|
https://en.wikipedia.org/wiki/Integral_quaternion
|
That is, either a, b, c, d are all integers, or they are all half-integers. H is closed under quaternion multiplication and addition, which makes it a subring of the ring of all quaternions H. Hurwitz quaternions were introduced by Adolf Hurwitz (1919). A Lipschitz quaternion (or Lipschitz integer) is a quaternion whose components are all integers. The set of all Lipschitz quaternions L = { a + b i + c j + d k ∈ H ∣ a , b , c , d ∈ Z } {\displaystyle L=\left\{a+bi+cj+dk\in \mathbb {H} \mid a,b,c,d\in \mathbb {Z} \right\}} forms a subring of the Hurwitz quaternions H. Hurwitz integers have the advantage over Lipschitz integers that it is possible to perform Euclidean division on them, obtaining a small remainder. Both the Hurwitz and Lipschitz quaternions are examples of noncommutative domains which are not division rings.
|
https://en.wikipedia.org/wiki/Integral_quaternion
|
In mathematics, a Inoue–Hirzebruch surface is a complex surface with no meromorphic functions introduced by Inoue (1977). They have Kodaira dimension κ = −∞, and are non-algebraic surfaces of class VII with positive second Betti number. Sankaran (1987) studied some higher-dimensional analogues.
|
https://en.wikipedia.org/wiki/Inoue-Hirzebruch_surface
|
In mathematics, a J-structure is an algebraic structure over a field related to a Jordan algebra. The concept was introduced by Springer (1973) to develop a theory of Jordan algebras using linear algebraic groups and axioms taking the Jordan inversion as basic operation and Hua's identity as a basic relation. There is a classification of simple structures deriving from the classification of semisimple algebraic groups. Over fields of characteristic not equal to 2, the theory of J-structures is essentially the same as that of Jordan algebras.
|
https://en.wikipedia.org/wiki/J-structure
|
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.
|
https://en.wikipedia.org/wiki/Jackson_q-Bessel_function
|
In mathematics, a Jacobi form is an automorphic form on the Jacobi group, which is the semidirect product of the symplectic group Sp(n;R) and the Heisenberg group H R ( n , h ) {\displaystyle H_{R}^{(n,h)}} . The theory was first systematically studied by Eichler & Zagier (1985).
|
https://en.wikipedia.org/wiki/Jacobi_form
|
In mathematics, a Jacobi sum is a type of character sum formed with Dirichlet characters. Simple examples would be Jacobi sums J(χ, ψ) for Dirichlet characters χ, ψ modulo a prime number p, defined by J ( χ , ψ ) = ∑ χ ( a ) ψ ( 1 − a ) , {\displaystyle J(\chi ,\psi )=\sum \chi (a)\psi (1-a)\,,} where the summation runs over all residues a = 2, 3, ..., p − 1 mod p (for which neither a nor 1 − a is 0). Jacobi sums are the analogues for finite fields of the beta function. Such sums were introduced by C. G. J. Jacobi early in the nineteenth century in connection with the theory of cyclotomy.
|
https://en.wikipedia.org/wiki/Jacobi_sum
|
Jacobi sums J can be factored generically into products of powers of Gauss sums g. For example, when the character χψ is nontrivial, J ( χ , ψ ) = g ( χ ) g ( ψ ) g ( χ ψ ) , {\displaystyle J(\chi ,\psi )={\frac {g(\chi )g(\psi )}{g(\chi \psi )}}\,,} analogous to the formula for the beta function in terms of gamma functions. Since the nontrivial Gauss sums g have absolute value p1⁄2, it follows that J(χ, ψ) also has absolute value p1⁄2 when the characters χψ, χ, ψ are nontrivial. Jacobi sums J lie in smaller cyclotomic fields than do the nontrivial Gauss sums g. The summands of J(χ, ψ) for example involve no pth root of unity, but rather involve just values which lie in the cyclotomic field of (p − 1)th roots of unity.
|
https://en.wikipedia.org/wiki/Jacobi_sum
|
Like Gauss sums, Jacobi sums have known prime ideal factorisations in their cyclotomic fields; see Stickelberger's theorem. When χ is the Legendre symbol, J ( χ , χ ) = − χ ( − 1 ) = ( − 1 ) p + 1 2 .
|
https://en.wikipedia.org/wiki/Jacobi_sum
|
{\displaystyle J(\chi ,\chi )=-\chi (-1)=(-1)^{\frac {p+1}{2}}\,.} In general the values of Jacobi sums occur in relation with the local zeta-functions of diagonal forms. The result on the Legendre symbol amounts to the formula p + 1 for the number of points on a conic section that is a projective line over the field of p elements.
|
https://en.wikipedia.org/wiki/Jacobi_sum
|
A paper of André Weil from 1949 very much revived the subject. Indeed, through the Hasse–Davenport relation of the late 20th century, the formal properties of powers of Gauss sums had become current once more. As well as pointing out the possibility of writing down local zeta-functions for diagonal hypersurfaces by means of general Jacobi sums, Weil (1952) demonstrated the properties of Jacobi sums as Hecke characters.
|
https://en.wikipedia.org/wiki/Jacobi_sum
|
This was to become important once the complex multiplication of abelian varieties became established. The Hecke characters in question were exactly those one needs to express the Hasse–Weil L-functions of the Fermat curves, for example. The exact conductors of these characters, a question Weil had left open, were determined in later work.
|
https://en.wikipedia.org/wiki/Jacobi_sum
|
In mathematics, a Jacobian, named for Carl Gustav Jacob Jacobi, may refer to: Jacobian matrix and determinant Jacobian elliptic functions Jacobian variety Intermediate Jacobian
|
https://en.wikipedia.org/wiki/Jacobian
|
In mathematics, a Jaffard ring is a type of ring, more general than a Noetherian ring, for which Krull dimension behaves as expected in polynomial extensions. They are named for Paul Jaffard who first studied them in 1960. Formally, a Jaffard ring is a ring R such that the polynomial ring dim R = n + dim R , {\displaystyle \dim R=n+\dim R,\,} where "dim" denotes Krull dimension.
|
https://en.wikipedia.org/wiki/Jaffard_ring
|
A Jaffard ring that is also an integral domain is called a Jaffard domain. The Jaffard property is satisfied by any Noetherian ring R, and examples of non-Noetherian rings might appear to be quite difficult to find, however they do arise naturally. For example, the ring of (all) algebraic integers, or more generally, any Prüfer domain. Another example is obtained by "pinching" formal power series at the origin along a subfield of infinite extension degree, such as the example given in 1953 by Abraham Seidenberg: the subring of Q ¯ ] {\displaystyle {\overline {\mathbf {Q} }}]} consisting of those formal power series whose constant term is rational.
|
https://en.wikipedia.org/wiki/Jaffard_ring
|
In mathematics, a Janet basis is a normal form for systems of linear homogeneous partial differential equations (PDEs) that removes the inherent arbitrariness of any such system. It was introduced in 1920 by Maurice Janet. It was first called the Janet basis by Fritz Schwarz in 1998.The left hand sides of such systems of equations may be considered as differential polynomials of a ring, and Janet's normal form as a special basis of the ideal that they generate.
|
https://en.wikipedia.org/wiki/Janet_basis
|
By abuse of language, this terminology will be applied both to the original system and the ideal of differential polynomials generated by the left hand sides. A Janet basis is the predecessor of a Gröbner basis introduced by Bruno Buchberger for polynomial ideals. In order to generate a Janet basis for any given system of linear PDEs a ranking of its derivatives must be provided; then the corresponding Janet basis is unique. If a system of linear PDEs is given in terms of a Janet basis its differential dimension may easily be determined; it is a measure for the degree of indeterminacy of its general solution. In order to generate a Loewy decomposition of a system of linear PDEs its Janet basis must be determined first.
|
https://en.wikipedia.org/wiki/Janet_basis
|
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.
|
https://en.wikipedia.org/wiki/Jónsson–Tarski_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.
|
https://en.wikipedia.org/wiki/Kac_algebra
|
In mathematics, a Kakeya set, or Besicovitch set, is a set of points in Euclidean space which contains a unit line segment in every direction. For instance, a disk of radius 1/2 in the Euclidean plane, or a ball of radius 1/2 in three-dimensional space, forms a Kakeya set. Much of the research in this area has studied the problem of how small such sets can be.
|
https://en.wikipedia.org/wiki/Kakeya_conjecture
|
Besicovitch showed that there are Besicovitch sets of measure zero. A Kakeya needle set (sometimes also known as a Kakeya set) is a (Besicovitch) set in the plane with a stronger property, that a unit line segment can be rotated continuously through 180 degrees within it, returning to its original position with reversed orientation. Again, the disk of radius 1/2 is an example of a Kakeya needle set.
|
https://en.wikipedia.org/wiki/Kakeya_conjecture
|
In mathematics, a Kato surface is a compact complex surface with positive first Betti number that has a global spherical shell. Kato (1978) showed that Kato surfaces have small analytic deformations that are the blowups of primary Hopf surfaces at a finite number of points. In particular they have an infinite cyclic fundamental group, and are never Kähler manifolds. Examples of Kato surfaces include Inoue-Hirzebruch surfaces and Enoki surfaces. The global spherical shell conjecture claims that all class VII surfaces with positive second Betti number are Kato surfaces.
|
https://en.wikipedia.org/wiki/Kato_surface
|
In mathematics, a Kempe chain is a device used mainly in the study of the four colour theorem. Intuitively, it is a connected chain of points on a graph with alternating colors.
|
https://en.wikipedia.org/wiki/Kempe_chain
|
In mathematics, a Killing tensor or Killing tensor field is a generalization of a Killing vector, for symmetric tensor fields instead of just vector fields. It is a concept in pseudo-Riemannian geometry, and is mainly used in the theory of general relativity. Killing tensors satisfy an equation similar to Killing's equation for Killing vectors.
|
https://en.wikipedia.org/wiki/Killing_tensor
|
Like Killing vectors, every Killing tensor corresponds to a quantity which is conserved along geodesics. However, unlike Killing vectors, which are associated with symmetries (isometries) of a manifold, Killing tensors generally lack such a direct geometric interpretation. Killing tensors are named after Wilhelm Killing.
|
https://en.wikipedia.org/wiki/Killing_tensor
|
In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric. Killing fields are the infinitesimal generators of isometries; that is, flows generated by Killing fields are continuous isometries of the manifold. More simply, the flow generates a symmetry, in the sense that moving each point of an object the same distance in the direction of the Killing vector will not distort distances on the object.
|
https://en.wikipedia.org/wiki/Killing_vectors
|
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.
|
https://en.wikipedia.org/wiki/Regular_algebra
|
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.
|
https://en.wikipedia.org/wiki/Klein_geometry
|
In mathematics, a Klein surface is a dianalytic manifold of complex dimension 1. Klein surfaces may have a boundary and need not be orientable. Klein surfaces generalize Riemann surfaces. While the latter are used to study algebraic curves over the complex numbers analytically, the former are used to study algebraic curves over the real numbers analytically.
|
https://en.wikipedia.org/wiki/Klein_surface
|
Klein surfaces were introduced by Felix Klein in 1882.A Klein surface is a surface (i.e., a differentiable manifold of real dimension 2) on which the notion of angle between two tangent vectors at a given point is well-defined, and so is the angle between two intersecting curves on the surface. These angles are in the range ; since the surface carries no notion of orientation, it is not possible to distinguish between the angles α and −α. (By contrast, on Riemann surfaces are oriented and angles in the range of (-π,π] can be meaningfully defined.) The length of curves, the area of submanifolds and the notion of geodesic are not defined on Klein surfaces. Two Klein surfaces X and Y are considered equivalent if there are conformal (i.e. angle-preserving but not necessarily orientation-preserving) differentiable maps f:X→Y and g:Y→X that map boundary to boundary and satisfy fg = idY and gf = idX.
|
https://en.wikipedia.org/wiki/Klein_surface
|
In mathematics, a Kleinian group is a discrete subgroup of the group of orientation-preserving isometries of hyperbolic 3-space H3. The latter, identifiable with PSL(2, C), is the quotient group of the 2 by 2 complex matrices of determinant 1 by their center, which consists of the identity matrix and its product by −1. PSL(2, C) has a natural representation as orientation-preserving conformal transformations of the Riemann sphere, and as orientation-preserving conformal transformations of the open unit ball B3 in R3. The group of Möbius transformations is also related as the non-orientation-preserving isometry group of H3, PGL(2, C). So, a Kleinian group can be regarded as a discrete subgroup acting on one of these spaces.
|
https://en.wikipedia.org/wiki/Kleinian_group
|
In mathematics, a Kleinian model is a model of a three-dimensional hyperbolic manifold N by the quotient space H 3 / Γ {\displaystyle \mathbb {H} ^{3}/\Gamma } where Γ {\displaystyle \Gamma } is a discrete subgroup of PSL(2,C). Here, the subgroup Γ {\displaystyle \Gamma } , a Kleinian group, is defined so that it is isomorphic to the fundamental group π 1 ( N ) {\displaystyle \pi _{1}(N)} of the surface N. Many authors use the terms Kleinian group and Kleinian model interchangeably, letting one stand for the other. The concept is named after Felix Klein. Many properties of Kleinian models are in direct analogy to those of Fuchsian models; however, overall, the theory is less well developed. A number of unsolved conjectures on Kleinian models are the analogs to theorems on Fuchsian models.
|
https://en.wikipedia.org/wiki/Kleinian_model
|
In mathematics, a Kline sphere characterization, named after John Robert Kline, is a topological characterization of a two-dimensional sphere in terms of what sort of subset separates it. Its proof was one of the first notable accomplishments of R. H. Bing; Bing gave an alternate proof using brick partitioning in his paper Complementary domains of continuous curves A simple closed curve in a two-dimensional sphere (for instance, its equator) separates the sphere into two pieces upon removal. If one removes a pair of points from a sphere, however, the remainder is connected. Kline's sphere characterization states that the converse is true: If a nondegenerate locally connected metric continuum is separated by any simple closed curve but by no pair of points, then it is a two-dimensional sphere.
|
https://en.wikipedia.org/wiki/Kline_sphere_characterization
|
In mathematics, a Kochanek–Bartels spline or Kochanek–Bartels curve is a cubic Hermite spline with tension, bias, and continuity parameters defined to change the behavior of the tangents. Given n + 1 knots, p0, ..., pn,to be interpolated with n cubic Hermite curve segments, for each curve we have a starting point pi and an ending point pi+1 with starting tangent di and ending tangent di+1 defined by d i = ( 1 − t ) ( 1 + b ) ( 1 + c ) 2 ( p i − p i − 1 ) + ( 1 − t ) ( 1 − b ) ( 1 − c ) 2 ( p i + 1 − p i ) {\displaystyle \mathbf {d} _{i}={\frac {(1-t)(1+b)(1+c)}{2}}(\mathbf {p} _{i}-\mathbf {p} _{i-1})+{\frac {(1-t)(1-b)(1-c)}{2}}(\mathbf {p} _{i+1}-\mathbf {p} _{i})} d i + 1 = ( 1 − t ) ( 1 + b ) ( 1 − c ) 2 ( p i + 1 − p i ) + ( 1 − t ) ( 1 − b ) ( 1 + c ) 2 ( p i + 2 − p i + 1 ) {\displaystyle \mathbf {d} _{i+1}={\frac {(1-t)(1+b)(1-c)}{2}}(\mathbf {p} _{i+1}-\mathbf {p} _{i})+{\frac {(1-t)(1-b)(1+c)}{2}}(\mathbf {p} _{i+2}-\mathbf {p} _{i+1})} where... Setting each parameter to zero would give a Catmull–Rom spline. The source code found here of Steve Noskowicz in 1996 actually describes the impact that each of these values has on the drawn curve: The code includes matrix summary needed to generate these splines in a BASIC dialect.
|
https://en.wikipedia.org/wiki/Kochanek–Bartels_spline
|
In mathematics, a Kodaira surface is a compact complex surface of Kodaira dimension 0 and odd first Betti number. The concept is named after Kunihiko Kodaira. These are never algebraic, though they have non-constant meromorphic functions. They are usually divided into two subtypes: primary Kodaira surfaces with trivial canonical bundle, and secondary Kodaira surfaces which are quotients of these by finite groups of orders 2, 3, 4, or 6, and which have non-trivial canonical bundles.
|
https://en.wikipedia.org/wiki/Kodaira_surface
|
The secondary Kodaira surfaces have the same relation to primary ones that Enriques surfaces have to K3 surfaces, or bielliptic surfaces have to abelian surfaces. Invariants: If the surface is the quotient of a primary Kodaira surface by a group of order k = 1,2,3,4,6, then the plurigenera Pn are 1 if n is divisible by k and 0 otherwise. Hodge diamond: Examples: Take a non-trivial line bundle over an elliptic curve, remove the zero section, then quotient out the fibers by Z acting as multiplication by powers of some complex number z. This gives a primary Kodaira surface.
|
https://en.wikipedia.org/wiki/Kodaira_surface
|
In mathematics, a Koecher–Maass series is a type of Dirichlet series that can be expressed as a Mellin transform of a Siegel modular form, generalizing Hecke's method of associating a Dirichlet series to a modular form using Mellin transforms. They were introduced by Koecher (1953) and Maass (1950).
|
https://en.wikipedia.org/wiki/Koecher–Maass_series
|
In mathematics, a Kolmogorov automorphism, K-automorphism, K-shift or K-system is an invertible, measure-preserving automorphism defined on a standard probability space that obeys Kolmogorov's zero–one law. All Bernoulli automorphisms are K-automorphisms (one says they have the K-property), but not vice versa. Many ergodic dynamical systems have been shown to have the K-property, although more recent research has shown that many of these are in fact Bernoulli automorphisms. Although the definition of the K-property seems reasonably general, it stands in sharp distinction to the Bernoulli automorphism. In particular, the Ornstein isomorphism theorem does not apply to K-systems, and so the entropy is not sufficient to classify such systems – there exist uncountably many non-isomorphic K-systems with the same entropy. In essence, the collection of K-systems is large, messy and uncategorized; whereas the B-automorphisms are 'completely' described by Ornstein theory.
|
https://en.wikipedia.org/wiki/Kolmogorov_system
|
In mathematics, a Koszul–Tate resolution or Koszul–Tate complex of the quotient ring R/M is a projective resolution of it as an R-module which also has a structure of a dg-algebra over R, where R is a commutative ring and M ⊂ R is an ideal. They were introduced by Tate (1957) as a generalization of the Koszul resolution for the quotient R/(x1, ...., xn) of R by a regular sequence of elements. Friedemann Brandt, Glenn Barnich, and Marc Henneaux (2000) used the Koszul–Tate resolution to calculate BRST cohomology. The differential of this complex is called the Koszul–Tate derivation or Koszul–Tate differential.
|
https://en.wikipedia.org/wiki/Koszul–Tate_resolution
|
In mathematics, a Lagrangian foliation or polarization is a foliation of a symplectic manifold, whose leaves are Lagrangian submanifolds. It is one of the steps involved in the geometric quantization of a square-integrable functions on a symplectic manifold.
|
https://en.wikipedia.org/wiki/Lagrangian_foliation
|
In mathematics, a Lagrangian system is a pair (Y, L), consisting of a smooth fiber bundle Y → X and a Lagrangian density L, which yields the Euler–Lagrange differential operator acting on sections of Y → X. In classical mechanics, many dynamical systems are Lagrangian systems. The configuration space of such a Lagrangian system is a fiber bundle Q → ℝ over the time axis ℝ. In particular, Q = ℝ × M if a reference frame is fixed. In classical field theory, all field systems are the Lagrangian ones.
|
https://en.wikipedia.org/wiki/Lagrangian_system
|
In mathematics, a Laguerre plane is one of the three types of Benz plane, which are the Möbius plane, Laguerre plane and Minkowski plane. Laguerre planes are named after the French mathematician Edmond Nicolas Laguerre. The classical Laguerre plane is an incidence structure that describes the incidence behaviour of the curves y = a x 2 + b x + c {\displaystyle y=ax^{2}+bx+c} , i.e. parabolas and lines, in the real affine plane. In order to simplify the structure, to any curve y = a x 2 + b x + c {\displaystyle y=ax^{2}+bx+c} the point ( ∞ , a ) {\displaystyle (\infty ,a)} is added. A further advantage of this completion is that the plane geometry of the completed parabolas/lines is isomorphic to the geometry of the plane sections of a cylinder (see below).
|
https://en.wikipedia.org/wiki/Laguerre_plane
|
In mathematics, a Lambert series, named for Johann Heinrich Lambert, is a series taking the form S ( q ) = ∑ n = 1 ∞ a n q n 1 − q n . {\displaystyle S(q)=\sum _{n=1}^{\infty }a_{n}{\frac {q^{n}}{1-q^{n}}}.} It can be resumed formally by expanding the denominator: S ( q ) = ∑ n = 1 ∞ a n ∑ k = 1 ∞ q n k = ∑ m = 1 ∞ b m q m {\displaystyle S(q)=\sum _{n=1}^{\infty }a_{n}\sum _{k=1}^{\infty }q^{nk}=\sum _{m=1}^{\infty }b_{m}q^{m}} where the coefficients of the new series are given by the Dirichlet convolution of an with the constant function 1(n) = 1: b m = ( a ∗ 1 ) ( m ) = ∑ n ∣ m a n . {\displaystyle b_{m}=(a*1)(m)=\sum _{n\mid m}a_{n}.\,} This series may be inverted by means of the Möbius inversion formula, and is an example of a Möbius transform.
|
https://en.wikipedia.org/wiki/Lambert_series
|
In mathematics, a Lamé function, or ellipsoidal harmonic function, is a solution of Lamé's equation, a second-order ordinary differential equation. It was introduced in the paper (Gabriel Lamé 1837). Lamé's equation appears in the method of separation of variables applied to the Laplace equation in elliptic coordinates. In some special cases solutions can be expressed in terms of polynomials called Lamé polynomials.
|
https://en.wikipedia.org/wiki/Lamé_function
|
In mathematics, a Lattès map is a rational map f = ΘLΘ−1 from the complex sphere to itself such that Θ is a holomorphic map from a complex torus to the complex sphere and L is an affine map z → az + b from the complex torus to itself. Lattès maps are named after French mathematician Samuel Lattès, who wrote about them in 1918.
|
https://en.wikipedia.org/wiki/Lattès_map
|
In mathematics, a Laurent polynomial (named after Pierre Alphonse Laurent) in one variable over a field F {\displaystyle \mathbb {F} } is a linear combination of positive and negative powers of the variable with coefficients in F {\displaystyle \mathbb {F} } . Laurent polynomials in X form a ring denoted F {\displaystyle \mathbb {F} } . They differ from ordinary polynomials in that they may have terms of negative degree. The construction of Laurent polynomials may be iterated, leading to the ring of Laurent polynomials in several variables. Laurent polynomials are of particular importance in the study of complex variables.
|
https://en.wikipedia.org/wiki/Laurent_polynomials
|
In mathematics, a Lawvere–Tierney topology is an analog of a Grothendieck topology for an arbitrary topos, used to construct a topos of sheaves. A Lawvere–Tierney topology is also sometimes also called a local operator or coverage or topology or geometric modality. They were introduced by William Lawvere (1971) and Myles Tierney.
|
https://en.wikipedia.org/wiki/Lawvere–Tierney_topology
|
In mathematics, a LeBrun manifold is a connected sum of copies of the complex projective plane, equipped with an explicit self-dual metric. Here, self-dual means that the Weyl tensor is its own Hodge star. The metric is determined by the choice of a finite collection of points in hyperbolic 3-space. These metrics were discovered by Claude LeBrun (1991), and named after LeBrun by Michael Atiyah and Edward Witten (2002).
|
https://en.wikipedia.org/wiki/Lebrun_manifold
|
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.
|
https://en.wikipedia.org/wiki/Leavitt_path_algebra
|
In mathematics, a Lefschetz manifold is a particular kind of symplectic manifold ( M 2 n , ω ) {\displaystyle (M^{2n},\omega )} , sharing a certain cohomological property with Kähler manifolds, that of satisfying the conclusion of the Hard Lefschetz theorem. More precisely, the strong Lefschetz property asks that for k ∈ { 1 , … , n } {\displaystyle k\in \{1,\ldots ,n\}} , the cup product ∪ : H n − k ( M , R ) → H n + k ( M , R ) {\displaystyle \cup \colon H^{n-k}(M,\mathbb {R} )\to H^{n+k}(M,\mathbb {R} )} be an isomorphism. The topology of these symplectic manifolds is severely constrained, for example their odd Betti numbers are even. This remark leads to numerous examples of symplectic manifolds which are not Kähler, the first historical example is due to William Thurston.
|
https://en.wikipedia.org/wiki/Lefschetz_manifold
|
In mathematics, a Lefschetz pencil is a construction in algebraic geometry considered by Solomon Lefschetz, used to analyse the algebraic topology of an algebraic variety V.
|
https://en.wikipedia.org/wiki/Lefschetz_pencil
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.