In mathematics , a duality , generally speaking, translates concepts, theorems or mathematical structures ... family of scalars. For instance, linear algebra duality corresponds in this way to bilinear maps from pairs of vector spaces to scalars, the duality between distribution mathematics distributions and the associated ... or polytopes are themselves dualitymathematics order theoretic duality order theoretic duals . Duality of polytopes and order theoretic duality are both Involution mathematics involution s the dual polytope ... . citation contribution III.19 Duality title The Princeton Companion to Mathematics pages 187 ... of an Involution mathematics involution operation if the dual of A is B , then the dual of B is A . As involutions sometimes have fixed point mathematics fixed points , the dual of A is sometimes ... contexts, duality has numerous meanings, and although it is a very pervasive and important concept in modern mathematics ref harvnb Kostrikin 2001 ref and an important general theme that has manifestations in almost every area of mathematics , ref name PCM187L harvnb Gowers 2008 loc p. 187, col. 1 ref there is no single universally agreed definition that unifies all concepts of duality ... a test function, and Poincar duality corresponds similarly to intersection number , viewed as a pairing ... 189, col. 2 ref Duality can also be seen as a functor , at least in the realm of vector spaces ... A particularly simple form of duality comes from order theory . The duality order theory dual of a poset ... s, and ideal order theory ideals and filter mathematics filters . A particular order reversal of this type ... are similarly dual. A partial order may be interpreted as a category mathematics category in which .... The order reversing duality of partial orders can be extended to the concept of a dual category ..., ref harvnb Artstein Avidan Milman 2007 ref ref harvnb Artstein Avidan Milman 2008 ref a duality ... up to some simple symmetries. If math mathcal T 1, mathcal T 2 math are two duality transforms then their Function ... more details
wiktionarypar dualityDuality may refer to MathematicsDualitymathematics , a mathematical concept Dual category theory , a formalization of mathematical dualityDuality order theory , a concept regarding binary relations Duality projective geometry , general principle of projective geometry S duality homotopy theory Duality optimization List of dualities Mathematics List of dualities Mathematics .... Science Electrical and mechanical Duality electrical circuits , regarding isomorphism of electrical circuits Duality mechanical engineering , regarding isomorphism of some mechanical laws Physics ... the Maldacena duality Dual resonance model Duality electricity and magnetism Englert Greenberger duality relation Holographic duality Kramers Wannier duality List of dualities Physics List of dualities physics Mirror symmetry Montonen Olive duality Mysterious duality String duality is a class of Symmetry physics symmetries S duality T duality U duality Wave particle duality Titles Film Duality film Duality film , a Star Wars fan film by Dave Macomber and Mark Thomas Music Duality Lisa Gerrard & Pieter Bourke album Duality Lisa Gerrard & Pieter Bourke album , a 1998 album by Lisa Gerrard and Pieter Bourke Duality song Duality song , a 2004 single & Grammy nominated song by metal band Slipknot Duality Ra album Duality Ra album , a 2005 album by the rock band Ra Duality , a 2007 single from alternative rock punk band Bayside band Bayside Other Duality, a large format audio mixing console by Solid State Logic Duality Console Solid State Logic Duality CoPs refers to the notion of a duality ... to singular and plural See also Double disambiguation Dual disambiguation Duality principle disambiguation List of dualities Philosophy, Mathematics, Physics and Engineering Nondualism Philosophy Triality Mathematics disambig Category Numbers Category Symbolism bg de Dualit t es Dualidad eu Dual fr Dualit it Duality nl Dualiteit ru Duality ... more details
This article is about S duality strong weak duality in physics. For the mathematical S duality Spanier Whitehead duality , see S duality homotopy theory . Unreferenced date December 2009 In theoretical physics , S duality also a strong weak duality is an equivalence of two quantum field theory quantum field theories or string theory string theories . An S duality transformation maps states and vacua with coupling constant g in one theory to states and vacua with coupling constant 1 g in the dual theory. This has permitted the use of perturbation theory quantum mechanics perturbation theory , normally useful only for weakly coupled theories with g 1, to also describe the strongly coupled g 1 regimes of string theory, by mapping them onto dual, weakly coupled regimes. In the case of four dimensional quantum field theories, S duality was understood by Ashoke Sen , Nathan Seiberg , and others. In this context, it usually exchanges the electric field electric and magnetic field s and the electrically charged particles with magnetic monopole s . See Montonen Olive duality , Seiberg duality . Many more examples come from string theory S duality relates type IIB string theory with the coupling constant g to the same type IIB string theory with the coupling constant 1 g . Similarly, type I string theory with the coupling g is equivalent to the SO 32 heterotic string theory with the coupling constant 1 g . Perhaps most amazing are the S dualities of type IIA string theory and E sub 8 sub ... dimension of size g . S duality has been rigorously shown to hold in some lattice model physics lattice ... model for U 1 , then its dual via the Kramers Wannier duality to a model where the vertices take on values ... . S duality typically exchanges local charges with topological charge s. See also T duality U duality Jordan Wigner transformation DEFAULTSORT S Duality Category Quantum field theory Category Statistical mechanics Category String theory Category Duality theories Category Lattice models String theory ... more details
T duality is a symmetry of quantum field theories with differing classical descriptions, ref cite web url http ncatlab.org nlab show T duality title nlab page on T duality ref of which the relationship between small and large distances in various string theory string theories is a special case. ref cite web url http www.staff.science.uu.nl caval101 homepage Research files t duality.pdf title Generalised complex geometry and T duality ref Discussion of the subject originated in a paper by T. S. Buscher and was further developed by Martin Rocek and Erik Verlinde . T duality is not present in ordinary particle physics particle theory , indicating that strings would experience spacetime in a way that is fundamentally distinct than the way particles do. It relates different string theory string theories that were thought to be unrelated before T duality was understood. T duality preceded the Second .... However, many of these geometries result in the same physics and this is the basis of T duality ... at radius math VAR &alpha VAR VAR R VAR . This relationship is an example of T duality. Bosonic String To illustrate the ideas of T duality, consider the bosonic string theory bosonic string ... The idea of T duality can be extended to more general backgrounds and even to superstring theory superstring theories . T duality interchanges the Type II string theory type II superstrings with each ... string wrapped once around the direction in question. Under T duality, it will be mapped to a IIB ... T duality acting on D branes changes their dimension by 1 or 1. Mirror Symmetry Andrew Strominger , Shing Tung Yau , and Eric Zaslow have showed that mirror symmetry can be understood as T duality applied ... also S duality U duality Mirror symmetry String theory Dualities Dualities in string theory References ... year 1987 citation first1 M. last1 Rocek first2 E. last2 Verlinde title Duality, quotients and currents journal Nuclear Phys. B, 373 3 630 646 year 1992 Category Duality theories Category String theory ... more details
Duality principle or principle of duality may refer to Duality projective geometry Duality order theory Duality principle Boolean algebra Duality principle for sets Duality principle optimization theory Lagrange dualityDuality principle in functional analysis, used in the large sieve method of analytic number theory Wave particle duality See also DualitymathematicsDuality disambiguation Dual disambiguation List of dualities mathdab ... more details
Multiple issues notability February 2012 technical February 2012 unreferenced January 2012 Mergeto Duality optimization date March 2012 In applied mathematics , weak duality is a concept in optimization which states that the solution to the primal minimization problem is always greater than or equal to the solution to an associated dual problem . This is opposed to strong duality which only holds in certain cases. Weak duality states that the duality gap is always greater than or equal to 0. Category Mathematical optimization Category Linear programming Category Convex optimization mathapplied stub ... more details
In mathematics , Lefschetz duality is a version of Poincar duality in geometric topology , applying to a manifold with boundary . Such a formulation was introduced by harvs txt authorlink Solomon Lefschetz last Lefschetz year 1926 , at the same time introducing relative homology , for application to the Lefschetz fixed point theorem . ref Biographical Memoirs By National Research Council Staff 1992 , p. 297. ref There are now numerous formulations of Lefschetz duality or Poincar Lefschetz duality , or Alexander Lefschetz duality . Formulations Let M be an orientable compact space compact manifold of dimension n , with boundary N , and let z be the fundamental class of M . Then cap product with z induces a pairing of the co homology group s of M and the relative co homology of the pair M , N and this gives rise to isomorphisms of H sup k sup M , N with H sub n k sub M , and of H sub k sub M , N with H sup n k sup M . ref James W. Vick, Homology Theory An Introduction to Algebraic Topology 1994 , p. 171. ref Here N can in fact be empty, so Poincar duality appears as a special case of Lefschetz duality. There is a version for triples. Let A and B denote two subspaces of the boundary N , themselves compact orientable manifolds with common boundary Z , which is the intersection of A and B . Then there is an isomorphism math D M H p M,A,Z to H n p M,B,Z . math Notes references References springer id Lefschetz duality Citation last1 Lefschetz first1 Solomon title Transformations of Manifolds with a Boundary jstor 84764 publisher National Academy of Sciences year 1926 journal Proceedings of the National Academy of Sciences Proceedings of the National Academy of Sciences of the United States of America issn 0027 8424 volume 12 issue 12 pages 737 739 Category Duality theories Category Manifolds ... more details
In mathematics, Tate duality or Poitou Tate duality is a duality theorem for Galois cohomology groups of modules over the Galois group of an algebraic number field or local field , introduced by harvs txt last Tate author link John Tate year 1962 and harvtxt Poitou 1967 . Local Tate duality main local Tate duality Local Tate duality says there is a perfect pairing of finite groups math displaystyle H r k,M times H 2 r k,M rightarrow H 2 k,G m Q Z math where M is a finite group scheme and M &prime its dual Hom M , G sub m sub . See also Artin Verdier duality Tate pairing References Citation last1 Haberland first1 Klaus title Galois cohomology of algebraic number fields url http books.google.com books?id PwbvAAAAMAAJ publisher VEB Deutscher Verlag der Wissenschaften id MR 519872 year 1978 Citation last1 Poitou first1 Georges title Cohomologie galoisienne des modules finis url http books.google.com books?id E3GmAAAAIAAJ publisher Dunod location Paris series S minaire de l Institut de Math matiques de Lille, sous la direction de G. Poitou. Travaux et Recherches Math matiques, id MR 0219591 year 1967 volume 13 chapter Propri t s globales des modules finis pages 255 277 Citation last1 Tate first1 John author1 link John Tate title Proceedings of the International Congress of Mathematicians Stockholm, 1962 url http mathunion.org ICM ICM1962.1 publisher Inst. Mittag Leffler location Djursholm id MR 0175892 year 1963 chapter Duality theorems in Galois cohomology over number fields pages 288 295 Category Algebraic number theory Category Galois theory ... more details
In mathematics, coherent duality is any of a number of generalisations of Serre duality , applying to coherent sheaves , in algebraic geometry and complex manifold theory, as well as some aspects of commutative ... expressed, with the advent of sheaf theory , in a way that made an analogy with Poincar duality ... of Jean Pierre Serre was extended to a proper morphism Serre duality was recovered as the case ... theory is now sometimes called Serre Grothendieck Verdier duality , and is a basic tool in algebraic geometry. A treatment of this theory, Residues and Duality 1966 by Robin Hartshorne , became an accessible ..., as for the versions of Poincar duality that are not for closed manifold s, requires some version ... local duality and subsequently. The 1992 Greenlees May duality is part of the continuing consideration of this area. Adjoint functor point of view Images of sheaves While Serre duality uses ... turn, Grothendieck reformulated general coherent duality as the existence of a right adjoint ... meaning, as the notation suggests, that duality is defined by integration. To be more precise ... a statement in more classical language, but still wider than Serre duality, Hartshorne Algebraic Geometry ... statement of Grothendieck duality for a projective or proper morphism math f X rightarrow Y math of noetherian schemes of finite dimension, found in Hartshorne Residues and duality is the following .... See also Verdier duality Notes references References Citation last1 Greenlees first1 J. P. C. last2 ... Hartshorne title Residues and Duality publisher Springer Verlag location Berlin, New York series Lecture Notes in Mathematics 20 year 1966 pages 20 48 Citation last1 Neeman first1 Amnon title The Grothendieck duality theorem via Bousfield s techniques and Brown representability doi 10.1090 S0894 0347 ... sheaves pages 393 408 DEFAULTSORT Coherent Duality Category Topological methods of algebraic geometry Category Sheaf theory Category Duality theories ... more details
In algebraic geometry , a branch of mathematics , Serre duality is a dualitymathematicsduality present on non singular projective algebraic variety algebraic varieties V of dimension n and in greater generality for vector bundle s and further, for coherent sheaf coherent sheaves . It shows that a cohomology group H sup i sup is the dual space of another one, H sup n &minus i sup . If the variety is defined over the complex number s, this yields different information from Poincar duality , which relates H sup i sup to H sup 2 n &minus i sup , considering V as a real manifold of dimension 2 n . In the case for holomorphic vector bundle E over a smooth compact complex manifold V , the statement is in the form math H q V,E cong H n q V,K otimes E ast ast , math in which V is not necessarily projective. Algebraic curve The case of algebraic curve s was already implicit in the Riemann Roch theorem . For a curve C the coherent groups H sup i sup vanish for i 1 but H sup 1 sup does enter implicitly ... D . That is, Serre duality in this case relates groups H sup 1 sup D and H sup 0 sup KD , and we are reading ... , in terms of the genus mathematics genus of the curve, which is h sup 1 sup C , O sub C sub , and the degree of D . It is this expression that can be generalised to higher dimensions. Serre duality ... sheaf K . The duality shows why these approaches coincide. Origin and generalisations The origin ... Grothendieck , Serre duality becomes a part of coherent duality in a much broader setting. While the role of K above in general Serre duality is played by the determinant line bundle of the cotangent ... title Duality Citation last1 Huybrechts first1 Daniel title Complex geometry publisher Springer Verlag ... duality for curves http rigtriv.wordpress.com 2008 05 29 serre duality Serre duality at the weblog ... connection between poincare duality and serre duality A link between Poincar and Serre dualities ... manifolds Category Duality theories Category Article Feedback 5 ... more details
citation style date April 2010 In mathematics , Esakia duality is the equivalence of categories dual equivalence between the category mathematics category of Heyting algebra s and the category of Esakia space s. Esakia duality provides an order topological representation of Heyting algebras via Esakia spaces. Let Esa denote the category of Esakia spaces and Esakia space Esakia morphisms Esakia morphisms . Let math H be a Heyting algebra, math X denote the set of prime ideal prime filters of math H , and math &le denote set theoretic inclusion on the prime filters of math H . Also, for each math a small &isin small H , let math &phi a x small &isin small X a small &isin small x , and let math &tau denote the topology on math X generated by math &phi a , X &minus &phi a a small &isin small H . Theorem ref Esakia 1974 . ref math X , &tau ,&le is an Esakia space, called the Esakia dual of math H . Moreover, math &phi is a Heyting algebra isomorphism from math H onto the Heyting algebra of all clopen set clopen up set s of math X , &tau ,&le . Furthermore, each Esakia space is isomorphic in Esa to the Esakia dual of some Heyting algebra. This representation of Heyting algebras by means of Esakia spaces is functorial and yields a dual equivalence between the category HA of Heyting algebras and Heyting algebra homomorphisms and the category Esa of Esakia spaces and Esakia morphisms. Theorem ref Esakia 1974 , Esakia 1985 , Bezhanishvili 2006 . ref HA is dually equivalent to Esa . Notes reflist References Esakia, L. 1974 . Topological Kripke models. Soviet Math. Dokl. , 15 147 151. Esakia, L. 1985 . Heyting Algebras I. Duality Theory Russian . Metsniereba, Tbilisi. Bezhanishvili, N. 2006 . Lattices of Intermediate and Cylindric Modal Logics . ILLC, University of Amsterdam. See also Duality theory for distributive lattices DEFAULTSORT Esakia Duality Category Topology Category Lattice theory ... more details
for duality over number fields Artin Verdier duality In mathematics, Verdier duality is a duality in sheaf theory that generalizes Poincar duality for manifold s. Verdier duality was introduced by harvs txt authorlink Jean Louis Verdier last Verdier year1 1967 year2 1995 as an analog for locally compact spaces of the coherent duality for schemes due to Grothendieck . It is commonly encountered when studying constructible or perverse sheaves . Verdier duality Verdier duality states that certain image functors for sheaves are actually adjoint functors . There are two versions. Global Verdier duality states that the higher direct image functor with compact supports Rf sub sub has a right adjoint ... Verdier duality states that math R , mathcal H om Rf mathcal F , mathcal G cong Rf ast R , mathcal ... global sections of both sides in the local statement gives global Verdier duality. The dualizing .... Part of what makes Verdier duality interesting in the singular setting is that when X is not a manifold ... F cong Rf D mathcal F math . li ul Poincar duality Poincar duality can be derived as a special case of Verdier duality. Here one explicitly calculates cohomology of a space using the machinery ... duality then states math Rp k X,k cong k X,p k . , math To understand how Poincar duality is obtained ... c X I bullet X ,k H 0 c X k X vee . math For the other side of the Verdier duality statement above, we ... X sub replaced with the same sheaf placed in degree i we get the classical Poincar duality math H ... series Progress in Mathematics isbn 978 0 8176 3274 8 year 1984 Citation last1 Gelfand first1 Sergei ... York series Lecture notes in mathematics isbn 978 3 540 08248 4 year 1977 volume 589 pages xii 484 ... 3 540 03953 2 id MR 0230732 year 1967 chapter A duality theorem in the etale cohomology of schemes ... cohomologie des espaces localement compacts pages Exp. No. 300, 337 349 DEFAULTSORT Verdier Duality Category Topology Category Homological algebra Category Sheaf theory Category Duality theories ... more details
In physics , U duality short for unified duality ref S. Mizoguchi, http iopscience.iop.org 0264 9381 17 4 308 On discrete U duality in M theory , 2000. ref is a symmetry of string theory or M theory combining S duality and T duality transformations. The term is most often met in the context of the U duality symmetry group of M theory as defined on a particular background space topological manifold . This is the union of all the S duality and T duality available in that topology. The narrow meaning of the word U duality is one of those dualities that can be classified neither as an S duality, nor as a T duality a transformation that exchanges a large geometry of one theory with the strong coupling of another theory, for example. References reflist Category string theory Category Duality theories physics stub it U dualit ... more details
In mathematics , the Poincar duality theorem named after Henri Poincar , is a basic result on the structure of the homology mathematics homology and cohomology group mathematics group s of manifold s. It states ... duality holds for any coefficient ring, so long as one has taken an orientation with respect ... duality holds mod 2 without any assumption of orientation. History A form of Poincar duality was first ... gave a new proof in terms of dual triangulations. Poincar duality did not take on its modern form ... product cup and cap product s and formulated Poincar duality in these new terms. Modern formulation The modern statement of the Poincar duality theorem is in terms of homology and cohomology if M is a closed ... . Homology and cohomology groups are defined to be zero for negative degrees, so Poincar duality .... The fact that this is an isomorphism of chain complexes is a proof of Poincar Duality. Roughly ... in this section. Then there are bilinear operator bilinear maps which are duality pairing duality ... the transverse intersection number of z with y and denominator n . The statement that the pairings are duality ... is an application of Poincar Duality math H i M simeq H n i M math together with the Universal ... Q mathbb Z math . Thus, Poincar duality says that math fH i M math and math fH n i M math are isomorphic ... duality was used by Przytycki and Yasuhara to give an elementary homotopy and diffeomorphism classification ... year 2003 issue 1 ref Thom Isomorphism Formulation Poincar Duality is closely related to the Thom space ... linking form . This formulation of Poincar Duality has become quite popular ref citation last ... year 1998 isbn 3540620435 ref as it provides a means to define Poincar Duality for any homology theory ... k theory . Generalizations and related results The Poincar Lefschetz duality theorem is a generalisation for manifolds with boundary. In the non orientable case, taking into account the sheaf mathematics ... Twisted Poincar duality . Blanchfield duality is a version of Poincar duality which provides an isomorphism ... more details
About String duality other forms of dualityDuality disambiguation String duality is a class of symmetry in physics symmetries in physics that link different string theory string theories , theories which ... of point particle s. Before the so called duality revolution there were believed to be five distinct ... , meaning right moving and left moving strings differ, group symmetry is E8 mathematics E sub ... special orthogonal group SO 32 and E8 mathematics E sub 8 sub E sub 8 sub . The thinking was that out ... are related by a duality transformation, each observable of the first theory can be mapped in some ... different descriptions of the same phenomena. A simple example of a duality is the equivalence ... related. T duality main T duality Suppose we are in ten spacetime dimensions, which means we have ... momentum around the circle, because its momentum is linked to its wavelength see Wave particle duality ... distance scale with a small distance scale. This type of duality is called T duality . T duality relates ... theories. T duality also relates type I string type I superstring theory to both type IIA and type .... T duality can be formally described by taking the left moving field to minus itself, so that the sum and the difference are interchanged, leading to switching of momentum and winding. S duality main S duality M theory Every force has a coupling constant , which is a measure of its strength ... with a very small one. This symmetry is called S duality . If two string theories are related by S duality ... , but the theory with weak coupling can. So if the two theories are related by S duality, then we ... theories related by S duality are type I string type I superstring theory with heterotic ... as M theory . Unlike the T duality, however, S duality has not been proven to even a physics level ... most string theorists believe in its validity. See also Dilaton M theory S duality String physics String String theory T duality U duality References Unreferenced date February 2007 Category String ... more details
In mathematics , Alexander duality refers to a duality theory presaged by a result of 1915 by James Waddell Alexander II J. W. Alexander , and subsequently further developed, particularly by P. S. Alexandrov and Lev Pontryagin . It applies to the homology theory properties of the complement of a subspace X in Euclidean space , a sphere , or other manifold. Modern statement Let X be a compact space compact , locally contractible space locally contractible subspace of the sphere S of dimension n . Let Y be the complement of X in S . Then if H stands for reduced homology or reduced cohomology , with coefficients in a given abelian group , there is an isomorphism between H sub q sub Y and H sup n q 1 sup X . Note that we can drop local contractibility as part of the hypothesis, if we use ech cohomology , which is designed to deal with local pathologies. Alexander s 1915 result To go back to Alexander s original work, it is assumed that X is a simplicial complex . Alexander had little of the modern apparatus, and his result was only for the Betti number s, with coefficients taken modulo 2. What to expect comes from examples. For example the Clifford torus construction in the 3 sphere shows that the complement of a solid torus is another solid torus which will be open if the other is closed, but this doesn t affect its homology. Each of the solid tori is from the homotopy point of view a circle . If we just write down the Betti numbers 1, 1, 0, 0 of the circle up to H sub 3 sub , since we are in the 3 sphere , then reverse as 0, 0, 1, 1 and then shift one to the left to get 0, 1, 1, 0 there is a difficulty, since we are not getting what we started with. On the other hand the same procedure applied to the reduced Betti numbers, for which the initial Betti number is decremented ... title Alexander duality id A a011290 Further reading Ezra Miller, Bernd Sturmfels, Combinatorial Commutative Algebra 2005 , Ch. 5 Alexander Duality Category Algebraic topology Category Duality theories ... more details
. Dantzig s foreword to Nering and Tucker, 1993 See also DualitymathematicsDuality Relaxation approximation ...Cleanup reorganize date February 2012 Mergefrom Weak duality Strong duality date March 2012 In constrained optimization mathematics optimization , it is often possible to convert the primal problem i.e. ..., the Wolfe dual problem and the Fenchel s duality theorem Fenchel dual problem . The Lagrangian dual .... Their difference is called the duality gap . For convex optimization problems, the duality gap is zero ... problem. Duality principle linked from redirect Duality principle optimization theory In optimization theory, the duality principle states that optimization problems may be viewed from either of two ... book title Duality in Vector Optimization author1 Bo , Radu Ioan author2 Wanka, Gert author3 Grad, Sorin Mihai year 2009 publisher Springer isbn 9783642028854 ref The duality gap is the difference of the right ... interior point regularity conditions in convex optimization. Applications of the duality theory to enlargements ... Inc. pages 106 113 isbn 981 238 067 1 mr 1921556 issue J year 2002 location River Edge, NJ ref Duality gap main duality gap The duality gap is the difference between the values of any dual problem primal ... primal value, then the duality gap is equal to math p d math . This value is always greater than or equal to 0. The duality gap is zero if and only if strong duality holds. Otherwise the gap is strictly positive and weak duality holds. ref cite book title Techniques of Variational Analysis ... ref In computational optimization, another duality gap is often reported, which is the difference ... problem. This alternative duality gap quantifies the discrepancy between the value of a current ... closure , that is the function that has the epigraph mathematics epigraph that is the closed convex ... A. title Numerical optimization Theoretical and practical aspects url http www.springer.com mathematics ... first2 Claude chapter 14 Duality for Practitioners title Convex analysis and minimization algorithms ... more details
In optimization problem s in applied mathematics , the duality gap is the difference between the dual problem primal and dual solutions . If math d math is the optimal dual value and math p math is the optimal primal value then the duality gap is equal to math p d math . This value is always greater than or equal to 0. The duality gap is zero if and only if strong duality holds. Otherwise the gap is strictly positive and weak duality holds. ref cite book title Techniques of Variational Analysis last1 Borwein first1 Jonathan last2 Zhu first2 Qiji year 2005 publisher Springer isbn 978 1 4419 2026 3 ref In general given two dual pair s separated space separated locally convex space s math left X,X right math and math left Y,Y right math . Then given the function math f X to mathbb R cup infty math , we can define the primal problem by math inf x in X f x . , math If there are constraint conditions, these can be built in to the function math f math by letting math f f I mathrm constraints math where math I math is the Characteristic function convex analysis indicator function . Then let math F X times Y to mathbb R cup infty math be a perturbation function such that math F x,0 f x math . The duality gap is the difference given by math inf x in X F x,0 sup y in Y F 0,y math where math F math is the convex conjugate in both variables. ref name BWG cite book title Duality in Vector Optimization ... interior point regularity conditions in convex optimization. Applications of the duality theory ... , another duality gap is often reported, which is the difference in value between any dual solution and the value of a feasible but suboptimal iterate for the primal problem. This alternative duality ... that has the epigraph mathematics epigraph that is the closed convex hull of the original primal ... Theoretical and practical aspects url http www.springer.com mathematics applications book 978 ... book last1 Hiriart Urruty first1 Jean Baptiste last2 Lemar chal first2 Claude chapter 14 Duality ... more details
In mathematics , there is an ample supply of duality of categories categorical dualities between certain category theory categories of topological space s and categories of partially ordered set s. Today, these dualities are usually collected under the label Stone duality , since they form a natural generalization of Stone s representation theorem for Boolean algebras . These concepts are named in honor of Marshall Stone . Stone type dualities also provide the foundation for pointless topology and are exploited ... formal semantics . This article gives pointers to special cases of Stone duality and explains a very general instance thereof in detail. Overview of Stone type dualities Probably the most general duality which is classically referred to as Stone duality is the duality between the category Sob of sober ... constructions are characteristic for this kind of duality, and are detailed below. Now one can ... to these basic dualities. Duality of sober spaces and spatial locales This section motivates and explains one of the most basic constructions of Stone duality the duality between topological spaces which are sober and frames i.e. complete Heyting algebra s which are spatial . This classical piece of mathematics ... therefore be considered as graduate level mathematics. Some prior exposure to the basics of category theory is recommended, although a deep understanding of the concepts of adjunction and duality ... a locale, which indeed gives an example of a central construction for Stone type duality theorems ... inverse of p sup &minus 1 sup 0 given by p sup &minus 1 sup 1 is a filter mathematics completely ... unit and counit, respectively. The duality theorem The above adjunction is not an equivalence of the categories Top and Loc or, equivalently, a duality of Top and Frm . For this it is necessary that both ... P. T. Johnstone, Stone Spaces , Cambridge Studies in Advanced Mathematics 3, Cambridge University Press ... Press, 1989. ISBN 0 521 36062 5. Category Topology Category Order theory Category Duality theories ... more details
corresponding characters on Pr fer group their Pontryagin dual group In mathematics, specifically in harmonic analysis and the theory of topological group s, Pontryagin duality explains the general ... cyclic groups. The Pontryagin duality theorem itself states that locally compact groups identify naturally ... for the theory of locally compact abelian groups and their duality during his early mathematical ... groups by Egbert van Kampen in 1935 and Andr Weil in 1940. Introduction Pontryagin duality places ... p adic topology. The dual group If G is a locally compact abelian group, a Character mathematics ... to its dual. It follows that the adele ring adele s are self dual. The Pontryagin duality ... functor and the dualization functor are not naturally equivalent. Pontryagin duality and the Fourier ... group G is that it carries an essentially unique natural Measure mathematics measure , the Haar ... This section is linked from Discrete space One important application of Pontryagin duality is the following ... topology on G and does not need Pontryagin duality. One uses Pontryagin duality to prove the converses ... compact or abelian. One use made of Pontryagin duality between compact abelian groups and discrete ... space s a special case, for real and complex vector spaces . The duality interchanges the subcategories of discrete groups and compact group s. If R is a ring mathematics ring and G is a left R module mathematics module , the dual group G will become a right R module in this way we can also see ... s in LCA is changed by duality into its opposite ring Constructing new rings from given ... that has been found useful in category theory is called Tannaka Krein duality but this diverges ... Plancherel measure on G . There are analogues of duality theory for noncommutative groups, some .... If this mapping is an isomorphism, we say that G satisfies Pontryagin duality. This has been extended ... duality part I infinite products , Duke Math. J. 15 1948 649 658, and part II direct and inverse ... more details
Unreferenced date December 2006 Duality of structure is one of Anthony Giddens coined phrases and main propositions in his explanation of structuration theory . The basis of the duality lies in the relationship the human agency Agency has with the Structure . In the duality, the Agency has much more influence on its lived environment than past structuralist theory had granted. The key to Giddens explanation is his focus on the knowledgeability of the agent and the fact that the agency cannot exist or be analyzed separately from its structure. They can only exist as a duality. The structural properties which he calls modalities sociology modalities help illustrate the dimensions of the duality. The Structure has both Norm sociology rules and resources or constraints and enabling qualities. Language is often used to exemplify these modalities. The system of interaction includes in itself rules of the language such as syntax but also leaves room for interpretations or the creation of completely new words. The system of interaction is responsible for maintaining a certain standard of consistency in order for the language to make sense to both the speaker and the interpreter. At the same time, the completeness of the language can always be manipulated and changed through interaction by the agent. The phrase is used in Chomsky by John Lyons to illustrate the differences between human and animal communication. It is described there as two levels of grammatical structure. The first being syntactic and the second being phonemes . DEFAULTSORT Duality Of Structure Category Sociological theories Category Critical theory Socio stub ... more details
Merge to Duality optimization date March 2012 Strong duality is a concept in optimization such that the dual problem primal and dual solutions are equivalent. This is as opposed to weak duality such that the primal problem has optimal value greater than the dual problem. Strong duality holds if and only if the duality gap is equal to 0. There are many conditions for strong duality to hold such as math F F math where math F math is the perturbation function relating the primal and dual problems and math F math is the convex conjugate biconjugate of math F math Citation needed date January 2012 the primal problem is a linear optimization linear optimization problem Slater s condition for a convex optimization convex optimization problem . ref name borwein cite book last1 Borwein first1 Jonathan last2 Lewis first2 Adrian title Convex Analysis and Nonlinear Optimization Theory and Examples edition 2 year 2006 publisher Springer isbn 978 0 387 29570 1 ref ref name boyd cite book last1 Boyd first1 Stephen last2 Vandenberghe first2 Lieven title Convex Optimization publisher Cambridge University Press year 2004 isbn 978 0 521 83378 3 url http www.stanford.edu boyd cvxbook bv cvxbook.pdf format pdf accessdate October 3, 2011 ref References Reflist Category Mathematical optimization Category Linear programming Category Convex optimization mathapplied stub ... more details
Refimprove date July 2011 Infobox album See Wikipedia WikiProject Albums Name Planetary Duality Type studio Artist The Faceless Cover Planetary Duality.jpg Released November 11, 2008 Recorded 2008 Genre Technical death metal br Progressive metal Length 31 42 Label Sumerian Records Sumerian Producer Michael Keene Last album Akeldama The Faceless album Akeldama 2006 This album Planetary Duality br 2008 Album ratings rev1 Der Metal Krieger rev1Score 9 10 ref cite web url http dermetalkrieger.com 2012 02 15 the faceless planetary duality 2008 title The Faceless Planetary Duality publisher Der Metal Krieger date 2012 15 02 accessdate 2012 15 02 ref Planetary Duality is the second full length album by technical death metal band The Faceless . It was released on November 11, 2008 through Sumerian Records in Europe through Lifeforce Records . The lyrics follow a sci fi horror theme of a reptillian race known as the Illuminati controlling the world and are based on the book The Children of the Matrix by David Icke. The album debuted at number 119 on the Billboard 200 Billboard 200 selling around 5,600 copies in its first week of release. ref cite web url http www.roadrunnerrecords.com blabbermouth.net news.aspx?mode Article&newsitemID 109201 title The Faceless Planetary Duality First Week Sales Revealed date 2008 11 19 publisher Blabbermouth.net Blabbermouth.com accessdate 2009 01 10 ref Track listing Tracklist title1 Prison Born length1 1 59 title2 The Ancient Covenant length2 4 02 title3 Shape Shifters length3 0 44 title4 Coldly Calculated Design length4 3 41 title5 Xenochrist length5 5 01 title6 Sons of Belial length6 4 46 title7 Legion of the Serpent length7 4 27 title8 Planetary Duality I Hideous Revelation length8 1 34 title9 Planetary Duality II A Prophecies Fruition length9 ... Reflist External links Allmusic class album id r1445496 label Planetary Duality Category 2008 albums ... Olofsson es Planetary Duality fr Planetary Duality fi Planetary Duality ... more details
about the Slipknot band Slipknot single Duality disambiguation Infobox Single See Wikipedia WikiProject Songs Name Duality Cover Duality.jpg Artist Slipknot band Slipknot from Album Vol. 3 The Subliminal ... musician Paul Gray , Joey Jordison Producer Rick Rubin Last single My Plague br 2002 This single Duality br 2004 Next single Vermilion song Vermilion br 2004 Duality is a song by the American heavy metal ... structure The album version of Duality is four minutes and twelve seconds long, and the radio edit ... url yes Duality . Allmusic . Retrieved 2010 03 05 ref The song opens with lead Singing vocalist Corey ... singles, Duality , like all the songs on Vol. 3 , does not have Parental Advisory explicit content within ... interview. Allmusic said that Duality s lyrics aren t unique to Slipknot but described it as otherwise strong . ref name vol 3 review Stylus Magazine said Duality had a grindcore riff . ref cite ... metal classic . Release and reception Duality was originally released as a CD single on May 4, 2004 ... Single2 cite news title Slipknot To Release Duality Seven Inch Vinyl Single publisher Blabbermouth.net ... track listing. Duality reached band records of number five and six in the Hot Mainstream Rock .... ref name MV1 cite news title Report Slipknot s Duality Video Cost Nearly Half A Million To Produce ... Single Duality small Single Version small 3 33 Don t Get Close small Duality Demo small 3 45 Disasterpiece small Live small 5 25 Duality small Video small Promo CD Duality small Edit small 3 33 Duality small Album Version small 4 13 7 Vinyl Duality small Single Version small 3 33 Don t Get Close small Duality Demo small 3 45 Limited Edition CD Duality small Single Version small 3 33 Don t Get ... class album id r917332 pure url yes allmusic Duality Overview . Allmusic. Retrieved 2010 05 08 ... album id r1007970 pure url yes allmusic Duality Import CD& 93 Overview . Allmusic. Retrieved 2010 05 08 ref References Reflist colwidth 30em Slipknot DEFAULTSORT Duality Song Category Slipknot songs ... more details
This article is about Dualities in Communities of practice . For other uses of Duality, see Duality disambiguation . In the context of a Community of practice the notion of a duality is used to capture the idea of the tension between two opposing forces which become a driving force for change and creativity. Wenger Harv Wenger 1998 uses the concept of dualities to examine the forces that create and sustain a Community of Practice. He describes a duality thus ... a single conceptual unit that is formed by two inseparable and mutually constitutive elements whose inherent tensions and complementarity give the concept richness and dynamism Harv Wenger 1998 p 66 . Some compare the concept of a duality to that of Yin and Yang , i.e. two mutually defining opposites. gallery Image Duality CoP .png Figure 1 adapted from Harv Hildreth Kimble 2002 gallery The opposing entities in a duality need to be viewed from a perspective of balance rather than opposition. The term implies a dynamism, continual change and mutual adjustment as the tensions that are inherent in dualities can be both creative and constraining. Harv Wenger 1998 identifies four dualities that exist in Communities of Practice participation ... Reification The Participation Reification duality is concerned with meaning. Meaning is created ... reification duality has been the focus of particular interest in this field Harv Hildreth Kimble 2002 . Designed Emergent The Designed Emergent duality focuses on time and captures the tension ... participants the opportunity to re negotiate existing meaning. The Designed Emergent duality is often ... Scheckler 2003 Identification Negotiability The Identification Negotiability duality is concerned ... of a community as a whole. Thus, this duality serves to combine both power and belonging in the shaping of the community. Local Global The Local Global duality concerns how one CoP relates to another ... last1 Hildreth first1 Paul last2 Kimble first2 Chris year 2002 title The duality of knowledge journal ... more details
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