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List Of Dualities
Mathematics In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of ''A'' is ''B'', then the dual of ''B'' is ''A''. * Alexander duality * Alvis–Curtis duality * Artin–Verdier duality * Beta-dual space * Coherent duality * De Groot dual * Dual abelian variety * Dual basis in a field extension * Dual bundle * Dual curve * Dual (category theory) * Dual graph * Dual group * Dual object * Dual pair * Dual polygon * Dual polyhedron * Dual problem * Dual representation * Dual q-Hahn polynomials * Dual q-Krawtchouk polynomials * Dual space * Dual topology * Dual wavelet * Duality (optimization) * Duality (order theory) * Duality of stereotype spaces * Duality (projective geometry) * Duality theory for distributive lattices * Dualizing complex * Dualizing sheaf * Eckman ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Polygon
In geometry, polygons are associated into pairs called duals, where the vertices of one correspond to the edges of the other. Properties Regular polygons are self-dual. The dual of an isogonal (vertex-transitive) polygon is an isotoxal (edge-transitive) polygon. For example, the (isogonal) rectangle and (isotoxal) rhombus are duals. In a cyclic polygon, longer sides correspond to larger exterior angles in the dual (a tangential polygon), and shorter sides to smaller angles. Further, congruent sides in the original polygon yields congruent angles in the dual, and conversely. For example, the dual of a highly acute isosceles triangle is an obtuse isosceles triangle. In the Dorman Luke construction, each face of a dual polyhedron is the dual polygon of the corresponding vertex figure. Duality in quadrilaterals As an example of the side-angle duality of polygons we compare properties of the cyclic and tangential quadrilaterals.Michael de Villiers, ''Some Adventures in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duality Theory For Distributive Lattices
In mathematics, duality theory for distributive lattices provides three different (but closely related) representations of bounded distributive lattices via Priestley spaces, spectral spaces, and pairwise Stone spaces. This duality, which is originally also due to Marshall H. Stone, generalizes the well-known Stone duality between Stone spaces and Boolean algebras. Let be a bounded distributive lattice, and let denote the set of prime filters of . For each , let . Then is a spectral space, where the topology on is generated by . The spectral space is called the ''prime spectrum'' of . The map is a lattice isomorphism from onto the lattice of all compact open subsets of . In fact, each spectral space is homeomorphic to the prime spectrum of some bounded distributive lattice. Similarly, if and denotes the topology generated by , then is also a spectral space. Moreover, is a pairwise Stone space. The pairwise Stone space is called the ''bitopological dual'' of . Ea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duality (projective Geometry)
In geometry, a striking feature of projective planes is the symmetry of the roles played by points and lines in the definitions and theorems, and ( plane) duality is the formalization of this concept. There are two approaches to the subject of duality, one through language () and the other a more functional approach through special mappings. These are completely equivalent and either treatment has as its starting point the axiomatic version of the geometries under consideration. In the functional approach there is a map between related geometries that is called a ''duality''. Such a map can be constructed in many ways. The concept of plane duality readily extends to space duality and beyond that to duality in any finite-dimensional projective geometry. Principle of duality A projective plane may be defined axiomatically as an incidence structure, in terms of a set of ''points'', a set of ''lines'', and an incidence relation that determines which points lie on which lines. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stereotype Space
In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space (TVS) for which the canonical evaluation map from X into its bidual (which is the strong dual of the strong dual of X) is an isomorphism of TVSs. Since a normable TVS is reflexive if and only if it is semi-reflexive, every normed space (and so in particular, every Banach space) X is reflexive if and only if the canonical evaluation map from X into its bidual is surjective; in this case the normed space is necessarily also a Banach space. In 1951, R. C. James discovered a Banach space, now known as James' space, that is reflexive but is nevertheless isometrically isomorphic to its bidual (any such isomorphism is thus necessarily the canonical evaluation map). Reflexive spaces play an important role in the general theory of locally convex TVSs and in the theory of Banach spaces in particular. Hilbert spaces are prominent examples of reflexive Banach spaces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duality (order Theory)
In the mathematical area of order theory, every partially ordered set ''P'' gives rise to a dual (or opposite) partially ordered set which is often denoted by ''P''op or ''P''''d''. This dual order ''P''op is defined to be the same set, but with the inverse order, i.e. ''x'' ≤ ''y'' holds in ''P''op if and only if ''y'' ≤ ''x'' holds in ''P''. It is easy to see that this construction, which can be depicted by flipping the Hasse diagram for ''P'' upside down, will indeed yield a partially ordered set. In a broader sense, two partially ordered sets are also said to be duals if they are dually isomorphic, i.e. if one poset is order isomorphic to the dual of the other. The importance of this simple definition stems from the fact that every definition and theorem of order theory can readily be transferred to the dual order. Formally, this is captured by the Duality Principle for ordered sets: : If a given statement is valid for all partially ordered sets, then its dual statement, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duality (optimization)
In mathematical optimization theory, duality or the duality principle is the principle that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem. If the primal is a minimization problem then the dual is a maximization problem (and vice versa). Any feasible solution to the primal (minimization) problem is at least as large as any feasible solution to the dual (maximization) problem. Therefore, the solution to the primal is an upper bound to the solution of the dual, and the solution of the dual is a lower bound to the solution of the primal. This fact is called weak duality. In general, the optimal values of the primal and dual problems need not be equal. Their difference is called the duality gap. For convex optimization problems, the duality gap is zero under a constraint qualification condition. This fact is called strong duality. Dual problem Usually the term "dual problem" refers to the ''Lagrangian dual problem'' but other ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Wavelet
In mathematics, a dual wavelet is the dual to a wavelet. In general, the wavelet series generated by a square-integrable function will have a dual series, in the sense of the Riesz representation theorem. However, the dual series is not itself in general representable by a square-integrable function. Definition Given a square-integrable function \psi\in L^2(\mathbb), define the series \ by :\psi_(x) = 2^\psi(2^jx-k) for integers j,k\in \mathbb. Such a function is called an ''R''-function if the linear span of \ is dense in L^2(\mathbb), and if there exist positive constants ''A'', ''B'' with 0 |
Dual Topology
In functional analysis and related areas of mathematics a dual topology is a locally convex topology on a vector space that is induced by the continuous dual of the vector space, by means of the bilinear form (also called ''pairing'') associated with the dual pair. The different dual topologies for a given dual pair are characterized by the Mackey–Arens theorem. All locally convex topologies with their continuous dual are trivially a dual pair and the locally convex topology is a dual topology. Several topological properties depend only on the dual pair and not on the chosen dual topology and thus it is often possible to substitute a complicated dual topology by a simpler one. Definition Given a dual pair (X, Y, \langle , \rangle), a dual topology on X is a locally convex topology \tau so that :(X, \tau)' \simeq Y. Here (X, \tau)' denotes the continuous dual of (X,\tau) and (X, \tau)' \simeq Y means that there is a linear isomorphism :\Psi : Y \to (X, \tau)',\quad y \mapsto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by constants. The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the . When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals, called the ''continuous dual space''. Dual vector spaces find application in many branches of mathematics that use vector spaces, such as in tensor analysis with finite-dimensional vector spaces. When applied to vector spaces of functions (which are typically infinite-dimensional), dual spaces are used to describe measures, distributions, and Hilbert spaces. Consequently, the dual space is an important concept in functional analysis. Early terms for ''dual'' include ''polarer Raum'' ah ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Q-Krawtchouk Polynomials
In mathematics, the dual ''q''-Krawtchouk polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties. Definition The polynomials are given in terms of basic hypergeometric function In mathematics, basic hypergeometric series, or ''q''-hypergeometric series, are ''q''-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series ''x'n'' is called hy ...s by :K_n(\lambda(x);c,N, q)=_3\phi_2(q^,q^,cq^;q^,0, q;q),\quad n=0,1,2,...,N, :where \lambda(x)=q^+cq^. References * * *{{dlmf, id=18, title=Chapter 18: Orthogonal Polynomials, first=Tom H. , last=Koornwinder, first2=Roderick S. C., last2= Wong, first3=Roelof , last3=Koekoek, , first4=René F. , last4=Swarttouw Orthogonal polynomials Q-analogs Special hypergeometric functions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Q-Hahn Polynomials
In mathematics, the dual ''q''-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties. Definition The polynomials are given in terms of basic hypergeometric function In mathematics, basic hypergeometric series, or ''q''-hypergeometric series, are ''q''-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series ''x'n'' is called hy ...s. R_n(q^+\gamma\delta q^,\gamma,\delta,N, q)=_3\phi_2\left begin q^,q^,\gamma\delta q^\\ \gamma q,q^\end ;q,q\right\quad n=0,1,2,...,N References * * * * * Orthogonal polynomials Q-analogs Special hypergeometric functions {{analysis-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
