Encyclopedia results for Wave–particle duality
wiktionarypar dualityDuality may refer to In philosophy, logic, and psychology Dualism , a twofold division in several spiritual, religious, and philosophical doctrines Dualism philosophy of mind , where the body and mind are considered to be irreducibly distinct De Morgan s Laws , specifically the ability to generate the dual of any logical expression. In mathematics Duality mathematics , a mathematical concept Dual category theory , a formalization of mathematical dualityDuality projective geometry , general principle of projective geometry In science Wave particleduality , a concept in quantum mechanics Duality electrical circuits , regarding isomorphism of electrical circuits Duality electricity and magnetism , regarding isomorphism of physical laws Physical dualities S duality T duality U duality S duality homotopy theory In film Duality Star Wars fan film Duality Star Wars fan film , a Star Wars fan film by Dave Macomber and Mark Thomas Duality film Duality film , a full length feature film created by Door Eleven Productions from Boston, Massachusetts In 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 in a Community of Practice Dual grammatical number grammatical number that some languages use in addition to singular and plural See also Nonduality Infinity Triality Duality principle disambiguation disambig Category Numbers Category Symbolism bg de Dualit t es Dualidad fr Dualit it Duality nl Dualiteit ru Duality ... more details
Unreferenced stub auto yes date December 2009 In theoretical physics , S duality also a strong weak duality is an equivalence of two quantum field theory quantum field theories , string theory string theories , or M theory . An S duality transformation maps the states and vacua with coupling constant math g math in one theory to states and vacua with coupling constant math 1 g math in the dual theory. This has permitted the use of perturbation theory quantum mechanics perturbation theory , normally useful only for weakly coupled theories with math g math less than 1, to also describe the strongly coupled math g math greater than 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 math g math to the same type IIB string theory with the coupling constant math 1 g math . Similarly, type I string theory with the coupling math g math is equivalent to the SO 32 heterotic string theory with the coupling constant math 1 g math . Perhaps ... g math . S duality has been rigorously shown to hold in some lattice model physics lattice models ... 1 , then its dual via the Kramers Wannier duality to a model where the vertices take on values in the dual ... gauge model with G as the gauge group with the electric and magnetic fields interchanged . S duality typically exchanges local charges with topological charge s. See also T duality U duality S duality homotopy theory Jordan Wigner transformation DEFAULTSORT S Duality Category Quantum field theory Category Statistical mechanics Category String theory Category Duality theories Category Lattice ... 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 experience spacetime in a way that is fundamentally ... 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. For example, consider the case where one of the dimensions is a circle of radius math VAR R VAR . A particle ... 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 compactified ... 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 ... wrapped once around the direction in question. Under T duality, it will be mapped to a IIB string ... to a IIB string with two units of momentum, and so on. Open Strings and D branes T duality ... Tung Yau , and Eric Zaslow have showed that mirror symmetry can be understood as T duality applied to three ... S duality U duality Mirror symmetry String theory Dualities Dualities in string theory References Becker ... 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 es Dualidad ... 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 particleduality See also Duality mathematics Duality disambiguation Dual disambiguation mathdab ... more details
Unreferenced date July 2009 U duality 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 and T dualities 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. Category string theory Category Duality theories physics stub it U dualit ... 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. DEFAULTSORT Duality Of Structure Category Sociological theories Category Critical theory Socio 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 Solomon Lefschetz in the 1920s, 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 External links http eom.springer.de P p073020.htm Category Duality theories Category Manifolds ... more details
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 , progressive metal Length 31 42 Label Sumerian Records Sumerian Producer Michael Keene Reviews Last album Akeldama The Faceless album Akeldama 2006 This album Planetary Duality br 2008 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 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 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 5 28 Personnel Michael Keene lead guitar , clean vocals , vocoder Steve Jones rhythm guitar Brandon Giffin bass guitar bass Derek Demon Carcass Rydquist singing vocals Lyle Cooper drum kit drums Additional musicians br Matthew Blackmar keyboard instrument keyboards Other P r Olofsson album cover cover artwork References reflist External links Allmusic class album id r1445496 Category 2008 albums Category The Faceless albums Category Sumerian Records albums fr Planetary Duality fi Planetary Duality ... more details
Unreferenced stub auto yes date December 2009 In theoretical physics , mysterious duality refers to similarities between M theory and del Pezzo surface s. More specifically there is a set of mathematical similarities between objects and laws describing M theory on k dimensional torus tori i.e. type II superstring theory on T sup k 1 sup for k 0 on one side, and geometry of del Pezzo surfaces for example, the cubic surface s on the other side. The main observation is that the large diffeomorphism s of del Pezzo surfaces match the Weyl group of the U duality group of the corresponding compactification of M theory. The elements of the second homology of the del Pezzo surfaces are mapped to various BPS objects of different dimensions in M theory . The complex projective plane P sup 2 sup C is related to M theory in 11 dimensions. When k points are blown up, the del Pezzo surface describes M theory on a k torus, and the exceptional del Pezzo surface, namely P sup 1 sup C × P sup 1 sup C , is connected with type IIB string theory in 10 dimensions. This conjecture was developed by Cumrun Vafa , Amer Iqbal , and Andrew Neitzke from Harvard University . DEFAULTSORT Mysterious Duality Category String theory Category Algebraic geometry Category Duality theories String theory stub ... 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
In mathematics , a duality , generally speaking, translates concepts, theorems or mathematical structures ... contexts, duality has numerous meanings, and although it is a very pervasive and important ... 187, col. 1 ref there is no single universally agreed definition that unifies all concepts of duality ... 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 ... a test function, and Poincar duality corresponds similarly to intersection number , viewed as a pairing ... 189, col. 2 ref Order reversing dualities A particularly simple form of duality comes from order theory . The duality order theory dual of a poset P X , is the poset P sup d sup X , comprising the same ... if and only if x y in the partial order. The order reversing duality of partial orders ... Artstein Avidan Milman 2008 ref a duality transform is just an involutive antiautomorphism math ..., mathcal T 2 math are two duality transforms then their Function composition composition is an Order isomorphism order automorphism of S thus, any two duality transforms differ only by an order automorphism .... A classical example of this is the duality of the platonic solid s, in which the cube and the octahedron .... The incidence preserving nature of the duality is reflected in the fact that the face lattice s of the primal and dual polyhedra or polytopes are themselves duality mathematics order theoretic duality order theoretic duals . Duality of polytopes and order theoretic duality are both Involution mathematics ... and with one edge for every two adjacent faces. The same concept of planar graph duality may be generalized ... from computational geometry the duality for any finite set S of points in the plane between the Delaunay triangulation of S and the Voronoi diagram of S . As with dual polyhedra and dual polytopes, the duality ... dual graphs. Matroid duality is an algebraic extension of planar graph duality, in the sense ... more details
In algebraic geometry , a branch of mathematics , Serre duality is a duality mathematics duality 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. In fact, the basic relation of the theorem involves l D and l K &minus D , where D is a divisor and K is a divisor of the canonical class . After Jean Pierre Serre Serre we recognise l K &minus ... D . That is, Serre duality in this case relates groups H sup 0 sup D and H sup 1 sup KD , and we are reading ... 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 Category Topological methods of algebraic geometry Category Complex manifolds Category Duality theories ... more details
Coherent duality in mathematics refers to 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 algebra that are part of the local theory. The historical roots of the theory lie in the idea ... 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 ... Robin author1 link Robin Hartshorne title Residues and Duality publisher Springer Verlag location ... first1 Amnon title The Grothendieck duality theorem via Bousfield s techniques and Brown representability ... Coherent Duality Category Topological methods of algebraic geometry Category Sheaf theory Category Duality theories ... more details
In quantum field theory , Seiberg duality , conjectured by Nathan Seiberg , is an S duality relating two different super QCD supersymmetric QCDs . The two theories are not identical, but they agree at low energies. More precisely under a renormalization group flow they flow to the same IR fixed point , and so are in the same universality class . It was first presented in Seiberg s 1994 article http arxiv.org abs hep th 9411149 Electric Magnetic Duality in Supersymmetric Non Abelian Gauge Theories ... of Montonen Olive duality in N 4 theories and electromagnetic duality in abelian group abelian theories. The statement of Seiberg duality Seiberg duality is an equivalence of the IR fixed point s in an N ... integers satisfying math N f N c 1 math . A stronger version of the duality relates not only the chiral ... W alpha M tilde Q c tilde Q math . Relations between the original and dual theories Being an S duality, Seiberg duality relates the strong coupling regime with the weak coupling regime , and interchanges ... . The meson s and baryon s are preserved by the duality. However in the electric theory the meson is a quark ... duality The moduli space s of the dual theories are identical. The global symmetries agree, as do the charges of the mesons and baryons. In certain cases it reduces to ordinary electromagnetic duality ... class. Six nontrivial anomalies may be computed on both sides of the duality, and they agree ... 2 N c 2 1 math math U 1 B 2 U 1 R math math 2 math math 2 math Generalizations Seiberg duality has been ... of Seiberg dualities known as a duality cascade . It was introduced by Matthew Strassler and Igor Klebanov in http arxiv.org abs hep th 0007191 Supergravity and a Confining Gauge Theory Duality Cascades and math chi math SB Resolution of Naked Singularities . It is not known whether Seiberg duality ... Simons theory Chern Simons terms. References http arxiv.org abs hep th 9411149 Electric Magnetic Duality ... Supersymmetry Category Quantum chromodynamics Category Duality theories ... 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 Mathematics Category Topology Category Lattice theory ... 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 ... , 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 music heavy metal ... History 5 1 21298.aspx ref Musical structure The album version of Duality is four minutes .... ref Allmusic class song id t7032073 pure url yes Duality . Allmusic . Retrieved 2010 03 05 ref The song ... to and features an alternative metal style. Unlike many previous Slipknot singles, Duality does ... 2010 03 05 ref Allmusic said that Duality s lyrics aren t unique to Slipknot. ref name vol 3 review Stylus Magazine said Duality had a grindcore riff. ref cite news date 2004 05 28 url http www.stylusmagazine.com ... journal Q magazine Q page 124 ref Release and reception Duality was originally released as a CD single .... ref name Single2 cite news title Slipknot To Release Duality Seven Inch Vinyl Single publisher ... includes the same track listing. Duality reached band records of number five and six in the Hot ... name MV1 cite news title Report Slipknot s Duality Video Cost Nearly Half A Million To Produce publisher ... 15 Track listings All songs written by Slipknot band Slipknot . CD Single Duality small Single Version small 3 33 Don t Get Close 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 3 45 Limited Edition CD Duality small Single Version small ... name allmusic Allmusic class album id r917332 pure url yes allmusic Duality Overview . Allmusic. Retrieved ... ref Allmusic class album id r1007970 pure url yes allmusic Duality Import CD Overview . Allmusic. Retrieved 2010 05 08 ref References Reflist colwidth 30em Slipknot DEFAULTSORT Duality Song Category ... more details
physics string s instead of point particle s. Before the so called duality revolution there were ... see Wave particleduality , and 2 R must be a multiple of that. In fact, the particle momentum around ...String duality is a class of symmetry in physics symmetries in physics that link different string theory ... strings major flaw a Particle physics particle with imaginary mass , called the tachyon , representing ... dualities. If two theories are related by a duality transformation, each observable of the first ... are two mathematically different descriptions of the same phenomena. A simple example of a duality is the equivalence of particle physics upon replacing matter with antimatter describing our universe ... of a physical system, in both classical field theory and quantum particle physics . But strings can ... theories end up being related. T duality main T duality Suppose we are in ten spacetime dimensions ... you around the circle and brings you back to where you started. A particle traveling around this circle ... exchanges a large distance scale with a small distance scale. This type of duality is called T duality . T duality relates type IIA string type IIA superstring theory to type IIB string type ... true for the two heterotic theories. T duality also relates type I string type I superstring theory ... number is related to their difference. T duality can be formally described by taking the left moving ... and winding. S duality main S duality M theory Every force has a coupling constant , which is a measure of its strength, and determines the chances of one particle to emit or absorb another particle. For electromagnetism , the coupling constant is proportional to the square of the electric ... t solve the whole theory exactly, because every particle may emit and absorb many other particles ... can happen in string theory. String theories have a coupling constant. But unlike in particle theories ... large coupling constant with a very small one. This symmetry is called S duality . If two string theories ... more details
In mathematics , the Poincar duality theorem, named after Henri Poincar , is a basic result on the structure ... Poincar 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 Rudyak, Y. On Thom ... Duality for any homology theory generalized homology theories provided one has a Thom Isomorphism ... duality theorem is a generalisation for manifolds with boundary. In the non orientable case, taking ... of orientability. Blanchfield duality is a version of Poincar duality which provides an isomorphism ..., there is a general Poincar duality theorem for homology theory generalized homology theories ... can be considered as the germinal idea for Poincar duality for generalized homology theories ... more details
In mathematics, Verdier duality is a generalization of the Poincar duality of manifold s to spaces with singularities. The theory was introduced by Jean Louis Verdier 1965 , and there is a coherent duality similar duality theory for schemes due to Grothendieck . It plays a role in the theory of perverse sheaves . Notation Images of sheaves F is a field mathematics field . The dimension of a locally compact space is the smallest integer n such that H sub c sub sup n 1 sup X , S 0 for all abelian sheaf mathematics sheaves S or infinity if no such integer n exists . X and Y are finite dimensional locally compact spaces, and f is a continuous map from X to Y . A , B is the set of morphisms between elements A and B of the derived category of sheaves on a space. f sub &lowast sub and f sup &lowast sup are the usual direct image functor direct and inverse image functor s between sheaves induced by f . The functor f sup &lowast sup is the left adjoint functor adjoint of f sub sub . f sub sub is the direct image with compact support and R f sup sup is the exceptional inverse image functor ... f shriek or f lower shriek and f upper shriek see also shriek map . Verdier duality Global Verdier duality states that Rf sub sub has a right adjoint f sup sup in the derived category, in other words ... by a complex of sheaves on the derived category. Local Verdier duality states that math R , mathrm ... over X . Taking homology of both sides gives global Verdier duality. The dualizing complex D sub ... D sup 2 sup S is isomorphic to S when S has constructible cohomology. li li Verdier duality If f is a continuous ... of dual complexes are dual. Poincar duality Poincar duality is a special case of Verdier duality ... F &minus n . Ordinary Poincar duality of a manifold can then be interpreted as the perfect pairing .... 300 1965 66 DEFAULTSORT Verdier Duality Category Topology Category Homological algebra Category Sheaf theory Category Duality theories ... 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
Infobox film name Duality image Duality title.jpg director Mark Thomas br Dave Macomber starring Dave Macomber br Mark Thomas br Stephen Muraoka music John Williams br Alan Derian cinematography Kevin Jones studio Crew of Two distributor TheForce.net released Film date 2001 2 10 Location runtime 6 15 country USA language English budget 3,000 Duality is a Star Wars fan film created by Mark Thomas and Dave Macomber that made its debut on the internet on February 10, 2001. It is one of the first fan films to exclusively use bluescreen footage composited onto virtual backgrounds. ref name Apple Cite web last Eaton first Nancy title Duality A Home made Sci fi Epic Film publisher Apple Inc. Apple.com date June 2001 url http web.archive.org web 20011217112524 www.apple.com hotnews articles 2001 06 duality accessdate March 8, 2011 ref Plot Sith apprentice Lord Rive Macomber , is tasked by his master, Darth Oz Thomas , to battle another apprentice, Darth Blight Muraoka , as a final test determining who would serve by Oz s side. The two would be Sith clash violently with their double bladed lightsabers while Oz watches via hologram. After a long and vicious lightsaber duel, Rive s weapon is severed in two and Blight loses his saber. A quick brawl later and the two engage in a final bout with the two halves of Rive s saber. It ends with Rive impaled and Blight bisected through the waist. After they die, Darth Oz appears in the flesh. Consumed with fury and wrath at his failure to procure a permanent apprentice, he electrocutes the bodies of the two failed apprentices with Force Lightning ... 2001 feats duality duality.htm accessdate March 8, 2011 ref Post production was split between Thomas ... Derian. Reception Reaction to Duality was mostly positive, generating over over a half million downloads ... Duality http www.youtube.com watch?v E03vPU Hsiw Duality on Youtube http www.ifilm.com ifilmdetail 1011883 Duality on ifilm Non canon Star Wars DEFAULTSORT Duality Star Wars Fan Film Category Star ... 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 in theoretical computer science for the study of 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 space s with continuous function s and the category SFrm of spatial complete Heyting algebra frames with appropriate frame homomorphisms. The dual category theory dual category of SFrm is the category of complete Heyting algebra locales denoted by SLoc . The equivalence of categories categorical equivalence of Sob and SLoc is the basis ... for this kind of duality, and are detailed below. Now one can easily obtain a number of other dualities ... lattice s. Many other Stone type dualities could be added to these basic dualities. Duality of sober ... of Stone duality the duality between topological spaces which are sober and frames i.e. complete ... a deep understanding of the concepts of adjunction and duality may well arise from examples such as the result ... of a central construction for Stone type duality theorems. Let us first look at the points of a topological ... transformation s &psi and &phi sup op sup provide the required unit and counit, respectively. The duality ..., a duality of Top and Frm . For this it is necessary that both &psi and &phi are isomorphisms in their respective ... Order theory Category Mathematical theorems Category Duality theories ... more details
In mathematics, specifically in harmonic analysis and the theory of topological group s, Pontryagin duality ... functor s, being represented respectively by K and T. Here is Ponryagin s duality theorem ... of Pontryagin duality is the following characterization of compact abelian topological groups ... of the compact open topology on G and does not need Pontryagin duality. One uses Pontryagin duality to prove the converses. The Bohr compactification is defined for any topological group G , regardless of whether G is locally compact or abelian. One use made of Pontryagin duality between ... dual of finite dimensional vector space s a special case, for real and complex vector spaces . The duality ... R modules. The ring End G of endomorphism s in LCA is changed by duality into its opposite ring ... groups and their duality were laid down by Lev Semenovich Pontryagin in 1934. His treatment relied on the group ... that has been found useful in category theory is called Tannaka Krein duality but this diverges ... Plancherel Plancherel measure on G . There are analogues of duality theory for noncommutative ... sense. If this mapping is an isomorphism, we say that G satisfies Pontryagin duality. This has been ... duality part I infinite products , Duke Math. J. 15 1948 649 658, and part II direct ... inverse limits of locally compact Hausdorff abelian groups satisfy Pontryagin duality. Note that an infinite ... Extensions of Pontryagin Duality , Math. Z. 143, 105 112 showed, among other facts, that every open subgroup of an abelian topological group which satisfies Pontryagin duality itself satisfies Pontryagin duality. More recently, S. Ardanza Trevijano and M.J. Chasco have extended the results of Kaplan mentioned above. They showed, in The Pontryagin duality of sequential limits of topological ... of sequences of abelian groups satisfying Pontryagin duality also satisfy Pontryagin duality if the groups ... if we want to consider Pontryagin duality beyond the locally compact case. In E. Martin Peinador ... more details
Infobox television episode Title Duel and Duality Series name Blackadder Image File Dual and Duality.jpg 200px Caption The duelling theme of the episode is illustrated by the titlecard artwork. Airdate 22 October 1987 Writer Ben Elton br Richard Curtis Director Mandie Fletcher Guests Stephen Fry br Gertan Klauber Episode list List of Blackadder episodes Series no 3 Episode 6 Prev Amy and Amiability Next Blackadder s Christmas Carol Duel and Duality is the sixth and final episode of the Blackadder the Third third series of the BBC sitcom Blackadder . Plot George Blackadder character Blackadder the Third Prince George has finally had a sexual encounter, but to Blackadder s astonishment, it emerges that it was with the two nieces of the Arthur Wellesley, 1st Duke of Wellington Duke of Wellington Stephen Fry . Blackadder warns the Prince that Wellington threatens to kill any who take sexual advantage of his relations. The Prince believes that Big Nose Wellington won t find out because he is still in Spain , Peninsular War fighting Napoleon I Napoleon Bonaparte . Unfortunately, he realizes that Wellington has triumphed six months ago and receives message that shows the Duke s intentions of challenging him to a duel . Horrified, the Prince enlists Mr. E. Blackadder Blackadder s help and Baldrick Blackadder the Third Baldrick suggests that the two change places, as Wellington does not know what the Prince looks like. Edmund isn t keen on the idea, but realizes that his mad Scotland Scottish cousin MacAdder also played by Rowan Atkinson , who has come down to London, could take his place. Later, Wellington decides to visit the Prince, and Blackadder and the Prince are forced to impersonate one another so that Wellington will not become suspicious during the actual duel. During Wellington s brief visit, Blackadder proves a far more competent Regent than the actual Prince Regent, and helps Wellington to mastermind the Battle of Trafalgar . The Prince proves less apt a butler than ... more details
In mathematics , Spanier&ndash Whitehead duality is a duality theory in homotopy theory , based on a geometrical idea that a topological space X may be considered as dual to its complement in the n sphere , where n is large enough. Its origins lie in the Alexander duality theory, in homology theory , concerning complements in manifold s. The theory is also referred to as S duality , but this can now cause possible confusion with the S duality of string theory . It is named for Edwin Spanier and J. H. C. Whitehead , who developed it in papers from 1955. The basic point is that sphere complements determine the homology, but not the homotopy type , in general. What is determined, however, is the stable homotopy type , which was conceived as a first approximation to homotopy type. Thus Spanier&ndash Whitehead duality fits into stable homotopy theory . References citation id MR 0056290 last Spanier first E. H. last2 Whitehead first2 J. H. C. title A first approximation to homotopy theory journal Proc. Nat. Acad. Sci. U.S.A. volume 39 year 1953 pages 655 660 citation id MR 0074823 last Spanier first E. H. last2 Whitehead first2 J. H. C. title Duality in homotopy theory. journal Mathematika volume 2 year 1955 pages 56 80 DEFAULTSORT Spanier Whitehead Duality Category Homotopy theory Category Duality theories ... more details
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