Encyclopedia results for Wave–particle duality
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 Semantics of programming languages 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 ... 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 ... 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 ... 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 ... Press, 1989. ISBN 0 521 36062 5. Category Topology Category Order theory Category Duality theories ... more details
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 ... 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 ... 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 ... 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 ... 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 Abelian ... 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 Captain Cook Blackadder Captain Cook 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 Prince finds someone else to take his place, as Wellington does not know what the Prince looks like. Blackadder prompts Baldrick to answer the Prince s objection that his face is known due to portraits hanging on every wall. Baldrick replies that his cousin told him that all portraits looked the same these days, because they were painted to a romantic ideal rather than the true depiction of the idiosyncratic facial qualities of the person in question . In a second reply, Baldrick suggests Blackadder as the one to fight the duel. Edmund isn t keen on the idea, but realizes that his mad Scotland Scottish cousin MacAdder also played ... 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 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 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
In theoretical physics , Montonen Olive duality is the oldest known example of S duality or a strong weak duality . It generalizes the electro magnetic symmetry of Maxwell s equations . It is named after Finland Finnish Claus Montonen and United Kingdom British David Olive . Overview In a four dimensional Yang Mills theory with extended supersymmetry N 4 supersymmetry , which is the case where the Montonen Olive duality applies, one obtains a physically equivalent theory if one replaces the gauge coupling constant g by 1 g . This also involves an interchange of the electrically charged particles and magnetic monopole s. See also Seiberg duality . In fact, there exists a larger modular group SL 2, Z symmetry where both g as well as theta angle are transformed non trivially. Mathematical formalism The gauge coupling and theta angle can be combined together to form one complex coupling math tau frac theta 2 pi frac 4 pi i g 2 . math Since the theta angle is periodic, there is a symmetry math tau mapsto tau 1. math The quantum mechanical theory with gauge group G but not the classical theory, except in the case when the G is abelian group abelian is also invariant under the symmetry math tau mapsto frac 1 n G tau math while the gauge group G is simultaneously replaced by its Langlands dual group sup L sup G and math n G math is an integer depending on the choice of gauge group. In the case the theta angle is 0, this reduces to the simple form of Montonen Olive duality stated above. References Edward Witten , http math.berkeley.edu index.php?module documents&JAS DocumentManager op viewDocument&JAS Document id 116 Notes from the 2006 Bowen Lectures , an overview of Electric Magnetic duality in gauge theory and its relation to the Langlands program Category Quantum field theory Category Duality theories quantum stub ... more details
In mathematics, Artin Verdier duality is a duality theorem for constructible abelian sheaves over the spectrum of a ring of algebraic numbers, introduced by harvs txt last1 Artin last2 Verdier year 1964 , that generalizes Tate duality . References Citation last1 Artin first1 Michael author1 link Michael Artin last2 Verdier first2 Jean Louis author2 link Jean Louis Verdier title Lecture notes prepared in connection with the seminars held at the summer institute on algebraic geometry. Whitney estate, Woods hole, Massachusetts. July 6 July 31 1964 url http www.jmilne.org math Documents woodshole3.pdf publisher American Mathematical Society location Providence, R.I. year 1964 chapter Seminar on tale cohomology of number fields Citation last1 Mazur first1 Barry author1 link Barry Mazur title Notes on tale cohomology of number fields url http www.numdam.org item?id ASENS 1973 4 6 4 521 0 id MathSciNet id 0344254 year 1973 journal Annales Scientifiques de l cole Normale Sup rieure. Quatri me S rie issn 0012 9593 volume 6 pages 521 552 Category Number theory ... more details
Technical date July 2011 In mathematics, the twisted Poincar duality is a theorem removing the restriction on Poincar duality to oriented manifold s. The existence of a global orientation is replaced by carrying along local information, by means of a local coefficient system . Integer valued formulation Let M be a d dimensional compact boundaryless differential manifold with orientation character w M . Then the cap product with the w twisted fundamental class induces Poincar duality isomorphisms between homology and cohomology math H M to H d M mathbb Z w math and math H M mathbb Z w to H d M math . Twisted Poincar duality for de Rham cohomology Another version of the theorem with real coefficients features the de Rham cohomology with values in the orientation bundle . This is the flat vector bundle flat real line bundle denoted math o M math , that is trivialized by coordinate charts of the manifold NM , with transition maps the sign of the Jacobian determinant of the charts transition maps. As a flat vector bundle flat line bundle , it has a de Rham cohomology, denoted by math H M mathbb R w math or math H M o M math . For M a compact manifold, the top degree cohomology is equipped with a so called trace morphism math theta H d M o M to mathbb R math , that is to be interpreted as integration on M , ie. evaluating against the fundamental class. The Poincar duality for differential forms is then the conjunction, for M connected, of the following two statements The trace morphism is a linear isomorphism, The cup product, or exterior product of differential forms math cup ... duality is contained in this statement, as understood from the fact that the orientation bundle ... vanishing parallel section. See also Local system Dualizing sheaf Verdier duality References ... duality the answers to this thread on MathOverflow The online book http www.maths.ed.ac.uk aar books ... Category Duality theories Category Theorems in topology ... more details
In mechanical engineering , many terms are associated into pairs called duals . A dual of a relationship is formed by interchanging force stress and deformation strain in an expression. Here is a partial list of mechanical dualities force &mdash deformation engineering deformation stress physics stress &mdash Strain materials science strain stiffness method &mdash flexibility method Examples Constitutive relation stress and strain Hooke s law . math sigma E varepsilon iff varepsilon frac 1 E sigma , math See also Duality electrical circuits References Fung, Y. C., A First Course in CONTINUUM MECHANICS , 2nd edition, Prentice Hall, Inc. 1977 Category Mechanical engineering Category Duality theories Mechanical engineering fa ... more details
File Canadian Duality Flag.svg 250px right thumb The flag adds blue lining stripes to the red flag of Canada to represent the unity of Canadian francophone s blue and English Canadian anglophones red . The Canadian Duality Flag also called the Canadian Unity Flag is an unofficial flag that was originally circulated to demonstrate the unity of Canada during the lead up to the 1995 Quebec referendum , at rallies for the no side. ref cite web accessdate 2008 04 13 url http www.crwflags.com fotw flags ca misc.html title Other Canadian flags Canada publisher CRW Flags ref Though the official national flag s colours are derived from British the red being from Saint George s Cross and French the white from the royal emblem used since King Charles VII of France Charles VII symbolism, ref cite web accessdate 2008 12 16 url http www.pch.gc.ca pgm ceem cced symbl df3 eng.cfm title Birth of the Canadian flag publisher Department of Canadian Heritage ref the Duality Flag design was chosen to explicitly represent the Francophone and English language Anglophone populations on the national flag by adding blue stripes to the red sections, roughly in proportion to the number of Canadians who are primarily French language French speaking. The blue was chosen as it is the main colour that is used on the flag of Quebec . ref cite web accessdate 2008 04 13 url http www.trcf.ca title Canadian Duality Flag publisher Canadian Duality ref Modified versions of the flag have been used to honour French Canadian ice hockey hockey players Maurice Richard Maurice The Rocket Richard and Bernie Geoffrion Bernie Boom Boom Geoffrion . In each case, the maple leaf was charged in white with the player s number 9 and 5 respectively . See also Flag of Canada References reflist Category 1995 in Canada Category Activism flags Category History of Quebec Category National symbols of Canada Category Politics of Canada Category Politics of Quebec Category Unofficial flags he ... more details
dablink For other meanings of the word not related to electrical circuits, see Duality disambiguation . In electrical engineering , electrical terms are associated into pairs called duals . A dual of a relationship is formed by interchanging voltage and current in an expression. The dual expression thus produced is of the same form, and the reason that the dual is always a valid statement can be traced to the Duality electricity and magnetism duality of electricity and magnetism . Here is a partial list of electrical dualities voltage &mdash Electric current current Series and parallel circuits parallel &mdash serial circuits Electrical resistance resistance &mdash Electrical conductance conductance Electrical impedance impedance &mdash admittance capacitance &mdash inductance Reactance electronics reactance &mdash susceptance short circuit &mdash open circuit Kirchhoff s current law &mdash Kirchhoff s voltage law . Th venin s theorem &mdash Norton s theorem History The use of duality in circuit theory is due to Alexander Russell who published his ideas in 1904. ref Belevitch, V, Summary of the history of circuit theory , Proceedings of the IRE , vol 50 , Iss 5, pp.848 855, May 1962 doi 10.1109 JRPROC.1962.288301 . ref ref Alexander Russell, A Treatise on the Theory of Alternating Currents , volume 1, chapter XXI, Cambridge University Press 1904 OCLC 264936988 . ref Examples Constitutive relations Resistor and conductor Ohm s law math v iR iff i vG , math Capacitor and inductor &ndash differential form math i C C frac d dt v C iff v L L frac d dt i L math Capacitor and inductor &ndash integral form math v C t V 0 1 over C int 0 t i C tau , d tau iff i L t I 0 1 over L int 0 t v L tau , d tau math Voltage division &mdash current division math v R 1 v frac R 1 R 1 R 2 iff ... Ls math math Z L Ls iff Y c Cs math See also Duality electricity and magnetism Duality mechanical engineering ... Library, Inc, New York, 1954, Chapter 6. Category Electrical engineering Category Duality theories ... more details
Infobox album See Wikipedia WikiProject Albums Name Duality Type Album Artist Ra U.S. band Ra Cover Duality Ra.jpg Released June 21, 2005 small United States U.S. small Recorded Henson Recording Studios, Hollywood, California Genre Hard rock Length 48 59 Label Universal Music Group Republic Records Producer Last album From One br 2002 This album Duality br 2005 Next album Raw Ra album Raw br 2006 Album ratings rev1 Antimusic rev1Score Rating 5 5 and rating 4.5 5 ref http www.antimusic.com features 05 radual.shtml ref rev2 Melodic.net rev2Score Rating 4 5 ref http www.melodic.net reviewsOne.asp?revnr 3238 ref rev3 UltimateGuitar rev3Score 9.3 10 ref http www.ultimate guitar.com reviews compact discs ra duality index.html ref Duality is a 2005 album from the band Ra U.S. band Ra . The style from this album has noticeably changed since Ra s debut album of From One . The album boasts a hard rock and melodic vibe whereas their first album had more of a nu metal feel. The middle eastern influences in the CD are still present, but are noticeably pushed back. Akin to their former album, Duality only claimed one single which was the song Fallen Angels . The tenth track Got Me Going was played on the radio in the background of the movie Hot Rod film Hot Rod . Janel Elizabeth has a cover of the song Swimming Upstream streamed on her Myspace . ref http www.myspace.com janelspage ref Early pressings of the album were immediately recalled from stores because of a manufacturing glitch, as skipping could be heard on track 2. The problem has since been corrected, and latest pressings of the album feature a sticker on the front of the case stating it to be a remastered version. Ra has sold 80,000 copies of Duality . ref http www.ra band.net history ref Duality also reached 137 on the Billboard 200, and also 2 for 18 weeks on Heatseekers Albums. ref http www.billboard.com charts rock songs album ra duality 720693 ref In the 2005 Melrock Awards, Duality was named the 3rd best modern rock ... more details
In mathematics, Alvis Curtis duality is a Duality mathematics duality operation on the Character mathematics characters of a reductive group over a finite field , introduced by harvs last Curtis first Charles W. authorlink Charles W. Curtis txt year 1980 and studied by his student harvs last Alvis first Dean txt year 1979 . harvs txt last Kawanaka year1 1981 year2 1982 introduced a similar duality operation for Lie algebras. Alvis Curtis duality has order 2 and is an isometry on generalized characters. harvtxt Carter 1985 loc 8.2 discusses Alvis Curtis duality in detail. Definition The dual of a character of a finite group G with a split BN pair is defined to be math zeta sum J subseteq R 1 J zeta G P J math Here the sum is over all subsets J of the set R of simple roots of the Coxeter system of G . The character su p G b P sub J sub is the truncation of to the parabolic subgroup P sub J sub of the subset J , given by restricting to P sub J sub and then taking the space of invariants of the unipotent radical of P sub J sub . The operation of truncation is the adjoint functor of parabolic induction . Examples The dual of the trivial character 1 is the Steinberg character . The dual of a Deligne Lusztig character R su b T p is sub G sub sub T sub R su b T p . The dual of a cuspidal character is 1 sup sup , where is the set of simple roots. The dual of the Gelfand Graev character is the character taking value Z sup F sup q sup l sup on the regular unipotent elements and vanishing elsewhere. References Citation last1 Alvis first1 Dean title The duality operation in the character ring of a finite Chevalley group doi 10.1090 S0273 0979 1979 14690 1 mr 546315 year 1979 journal American Mathematical Society. Bulletin. New Series issn 0002 9904 volume 1 issue ... Charles W. Curtis title Truncation and duality in the character ring of a finite group of Lie ... 9910 volume 69 issue 3 pages 411 435 Category Representation theory Category Duality theories ... more details
Unreferenced date December 2009 Dablink For other meanings of the word not related to electricity and magnetism, see Duality disambiguation . In physics, the electromagnetic dual concept is based on the idea that, in the static case, electromagnetism has two separate facets electric fields and magnetic field s. Expressions in one of these will have a directly analogous, or dual, expression in the other. The reason for this can ultimately be traced to special relativity where applying the Lorentz transformation to the electric field will transform it into a magnetic field. The electric field is the dual of the magnetic field . The electric displacement field is the dual of the magnetic field The H field magnetizing field . Faraday s law is the dual of Ampere s law . Gauss s law for electric field is the dual of Gauss s law for magnetism . The electric potential is the dual of the magnetic potential . Permittivity is the dual of Permeability electromagnetism permeability . Electrostriction is the dual of magnetostriction . Piezoelectricity is the dual of piezomagnetism . Ferroelectricity is the dual of ferromagnetism . An electrostatic motor is the dual of a electric motor magnetic motor Electret s are the dual of magnet permanent magnets The Faraday effect is the dual of the Kerr effect The Aharonov Casher effect is the dual to the Aharonov Bohm effect The magnetic monopole is the hypothetical dual of electric charge . See also Maxwell s equations Duality electrical circuits DEFAULTSORT Duality Electricity And Magnetism Category Electromagnetism ... more details
In mathematics , Tannaka Krein duality theory concerns the interaction of a compact group compact topological group and its category of linear representation s. Its natural extension to the non Abelian case is the Grothendieck duality Grothendieck duality theory . It extends an important mathematical duality between compact and discrete commutative topological groups, known as Pontryagin duality , to groups that are compact, but noncommutative . The theory is named for two men, the Soviet mathematician Mark Grigorievich Krein , and the Japanese Tadao Tannaka . In contrast to the case of commutative groups considered by Lev Pontryagin , the notion dual to a noncommutative compact group is not a group, but a category mathematics category G with some additional structures, formed by the finite dimensional representations of G . Duality theorems of Tannaka and Krein describe the converse passage from the category G back to the group G , allowing one to recover the group from its category of representations. Moreover, they in effect completely characterize all categories that can arise ... duality can be extended to the case of algebraic group s see tannakian category . Meanwhile, the original ... Krein duality category of representations of a group In Pontryagin duality theory for locally ... to Tannaka Krein duality theory was reawakened in the 1980s with the discovery of quantum group ... monoidal category . It turned out that a good duality theory of Tannaka Krein type also exists in this case ... duality theory for compact groups . Inventiones Mathematicae, 98 157 218, 1989. ref characterises ... http front.math.ucdavis.edu author Amini M UC Davis site with three articles on Tannaka Krein Duality ... 9507018 tannaka krein duality Quantum Principal Bundles and Tannaka Krein Duality by Mico Durdevic http ... An introduction to Tannaka duality and quantum groups , in Part II of Category Theory, Proceedings, Como ... Category Harmonic analysis Category Topological groups Category Duality theories ... more details
orphan date November 2009 Unreferenced date January 2007 In quantum information theory , the channel state duality refers to the correspondence between quantum channel s and quantum states described by density matrix density matrices . Phrased differently, the duality is the isomorphism between completely positive maps channels from A to C sup n × n sup , where A is a C algebra and C sup n × n sup denotes the n × n complex entries, and positive linear functionals state functional analysis state s on the tensor product math mathbb C n times n otimes A. math Details Let H sub 1 sub and H sub 2 sub be finite dimensional Hilbert spaces. The family of linear operators acting on H sub i sub will be denoted by L H sub i sub . Consider two quantum systems, indexed by 1 and 2, whose states are density matrices in L H sub i sub respectively. A quantum channel , in the Schr dinger picture, is a completely positive CP for short linear map math Phi L H 1 rightarrow L H 2 math that takes a state of system 1 to a state of system 2. Next we describe the dual state corresponding to . Let E sub i j sub denote the matrix unit whose ij th entry is 1 and zero elsewhere. The operator matrix math rho Phi Phi E ij ij in L H 1 otimes L H 2 math is called the Choi matrix of . By Choi s theorem on completely positive maps , is CP if and only if sub sub is positive semidefinite . One can view sub sub as a density matrix, and therefore the state dual to . The duality between channels and states refers to the map math Phi rightarrow rho Phi , math a linear bijection. This map is also called Jamio kowski isomorphism or Choi&ndash Jamio kowski isomorphism . DEFAULTSORT Channel State Duality Category Quantum information theory ... more details
In Galois cohomology , local Tate duality or simply local duality is a Duality mathematics duality for Galois module s for the absolute Galois group of a non archimedean local field . It is named after John Tate who first proved it. It shows that the duality mathematics dual of such a Galois module is the Tate twist of usual linear dual. This new dual is called the local Tate dual . Local duality combined with Tate s local Euler characteristic formula provide a versatile set of tools for computing the Galois cohomology of local fields. Statement Let K be a non archimedean local field, let K sup s sup denote a separable closure of K , and let G sub K sub Gal K sup s sup K be the absolute Galois group of K . Case of finite modules Denote by the Galois module of all roots of unity in K sup s sup . Given a finite G sub K sub module A of order prime to the characteristic algebra characteristic of K , the Tate dual of A is defined as math A prime mathrm Hom A, mu math i.e. it is the Tate twist of the usual dual A sup sup . Let H sup i sup K , A denote the group cohomology of G sub K sub with coefficients in A . The theorem states that the pairing math H i K,A times H 2 i K,A prime rightarrow H 2 K, mu mathbf Q mathbf Z math given by the cup product sets up a duality between H sup i sup K , A and H sup 2&minus i sup K , A sup &prime sup for i 0, 1, 2. ref harvnb Serre 2002 loc Theorem II.5.2 ref Since G sub K sub has cohomological dimension equal to two, the higher cohomology groups vanish. ref harvnb Serre 2002 loc II.4.3 ref Case of p adic representations Let p be a prime number . Let Q sub p sub 1 denote the cyclotomic character p adic cyclotomic ... group cohomology of G sub K sub with coefficients in V . Local Tate duality applied to V says ... Q p 1 mathbf Q p math which is a duality between H sup i sup K , V and H sup 2&minus i sup K ..., the higher cohomology groups vanish. See also Poitou Tate duality , a global version i.e. for global ... more details
Schur Weyl duality is a mathematical theorem in representation theory that relates irreducible finite dimensional representations of the general linear group general linear and symmetric group symmetric groups. It is named after two pioneers of representation theory of Lie group s, Issai Schur , who discovered the phenomenon, and Hermann Weyl , who popularized it in his books on quantum mechanics and classical groups as a way of classifying representations of unitary group unitary and general linear groups. Description Schur Weyl duality forms an archetypical situation in representation theory involving two kinds of symmetry that determine each other. Consider the tensor space math mathbb C n otimes mathbb C n otimes cdots otimes mathbb C n math with k factors. The symmetric group S sub k sub on k letters group action acts on this space on the left by permuting the factors, math sigma v 1 otimes v 2 otimes cdots otimes v k v sigma 1 1 otimes v sigma 1 2 otimes cdots otimes v sigma 1 k . math The general linear group GL sub n sub of invertible n × n matrices acts on it by the simultaneous matrix multiplication , math g v 1 otimes v 2 otimes cdots otimes v k gv 1 otimes gv 2 otimes cdots otimes gv k, quad g in GL n. math These two actions Equivariant map commute , and in its concrete form, the Schur Weyl duality asserts that under the joint action of the groups S sub k sub and GL sub n sub , the tensor space decomposes into a direct sum of tensor products of irreducible modules for these two groups that determine each other, math mathbb C n otimes mathbb C n otimes cdots otimes mathbb C n sum D pi k D otimes rho n D. math The summands are indexed by the Young diagram s D ... n sub . The abstract form of the Schur Weyl duality asserts that two algebras of operators on the tensor ... duality is the statement that the space of two tensors decomposes into symmetric and antisymmetric ... on invariant theory Schur duality, multiplicity free actions and beyond . The Schur lectures 1992 ... more details
The Kramers Wannier duality is a symmetry in statistical physics . It relates the Thermodynamic free energy free energy of a two dimensional square lattice Ising model at a low temperature to that of another Ising model at a high temperature. It was discovered by Hendrik Anthony Kramers Hendrik Kramers and Gregory Wannier in 1941. With the aid of this duality Kramers and Wannier found the exact location of the critical point thermodynamics critical point for the Ising model on the square lattice. Similar dualities establish relations between free energies of other statistical models. For instance, in 3 dimensions the Ising model is dual to an Ising gauge model. Intuitive idea The 2 dimensional Ising model exists on a lattice, which is a collection of squares in a chessboard pattern. With the finite lattice, the edges can be connected to form a torus. In theories of this kind, one constructs an Involution mathematics involutive transform . For instance, Lars Onsager suggested that the Star Triangle transformation could be used for the triangular lattice. ref Somendra M. Bhattacharjee, and Avinash Khare, Fifty Years of the Exact Solution of the Two Dimensional Ising Model by Onsager 1995 , arxiv cond mat 9511003 ref Now the dual of the discrete torus is dual lattice itself . Moreover, the dual of a highly disordered system high temperature is a well ordered system low temperature . This is because the fourier transform takes a high Bandwidth signal processing bandwidth signal more ... ref nonhomogenous torus, ref arXiv hep th 9703037, Duality of the 2D Nonhomogeneous Ising Model on the Torus ..., U. Grimm, R. J. Baxter ref lattices with twisted boundaries, ref arXiv hep th 0209048, Duality ... 0905.1924, Duality and Symmetry in Chiral Potts Model , Shi shyr Roan ref and many others. Derivation ... infty f N K,L kT lim N rightarrow infty frac 1 N log Z N K,L math the Kramers Wannier duality gives ... , implying sinh 2K sub c sub 1 , yielding kT sub c sub 2.2692J . See also Ising model S duality References ... more details
In mathematics , duality theory for distributive lattices provides three different but closely related representations of distributive lattice bounded distributive lattices via Priestley space s, spectral space s, and pairwise Stone space s. This generalizes the well known Stone duality between Stone space s and Boolean algebra structure Boolean algebra s. Let math L be a bounded distributive lattice, and let math X denote the set mathematics set of Ideal order theory prime filters of math L . For each math a small &isin small L , let math &phi sub sub a x small &isin small X a &isin x . Then math X , &tau sub sub is a spectral space, ref Stone 1937 , Johnstone 1982 ref where the topological space Definition topology math &tau sub sub on math X is generated by math &phi sub sub a a small &isin small L . The spectral space math X , &tau sub sub is called the prime spectrum of math L . The map mathematics map math &phi sub sub is a lattice isomorphism from math L onto the lattice of all compact set compact open set open subsets of math X , &tau sub sub . In fact, each spectral space is homeomorphism homeomorphic to the prime spectrum of some bounded distributive lattice. ref Stone 1937 , Johnstone 1982 ref Similarly, if math &phi sub &minus sub a x small &isin small X a ¬in x and math &tau sub &minus sub denotes the topology generated by math &phi sub &minus sub a a small &isin small L , then math X , &tau sub &minus sub is also a spectral space. Moreover, math X , &tau sub sub ..., all three are dually equivalent to Dist Duality for bounded distributive lattices Thus, there are three ... distributive lattices. See also Birkhoff s representation theorem Stone duality Stone s representation theorem for Boolean algebras Esakia duality Notes reflist References Priestley, H. A. 1970 ... duality. Technical Report CSR 06 13 , School of Computer Science, University of Birmingham. Bezhanishvili, G., Bezhanishvili, N., Gabelaia, D., Kurz, A. 2010 . Bitopological duality for distributive ... more details
and theorems, and plane duality is the formalization of this Metamathematics metamathematical concept. There are two approaches to the subject of duality, one through language the Principle of Duality Principle of Duality and the other a more functional approach. These are completely ... a duality . In specific examples, 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 details Incidence structure Dual structure If one defines a projective ... in the proof in C gives a statement of the proof in C . The Principle of Plane Duality says that dualizing ... concepts can be generalized to talk about space duality, where the terms points and planes are interchanged and lines remain lines . This leads to the Principle of Space Duality . Further generalization ... Duality as a mapping A plane duality is a map from a projective plane C P,L,I to its dual plane C L,P,I see Principle of Duality above which preserves incidence. That is, a plane duality will map ... duality which is an isomorphism is called a correlation . ref harvnb Dembowski 1968 pg.151. ref ... that the projective plane is of the Projective space PG 2, K type, with K a division ring, a duality .... This duality mapping concept can also be extended to higher dimensional spaces so the modifier plane can be dropped in those situations. Higher dimensional dualityDuality in the projective plane is a special case of duality for projective space s, transformations of PG n, K also denoted by K P ... correspond to planes, and lines correspond to lines. By restriction the dual polyhedron duality ... is the dual of the line. Duality mapping defined Given a line L in the projective plane, what ... red pair, one yellow pair, br and one blue pair. The duality is an isomorphism of incidence, so that, e.g. ... and lines one red pair, one yellow pair, br and one blue pair. The duality is an isomorphism of incidence ... more details
in the experiment, linked together as the Englert Greenberger duality relation . The next section will discuss the orthodox quantum mechanical interpretation of the duality relation in terms of wave particleduality. Of this experiment, Richard Feynman once said that it has in it the heart of quantum ... formulation of Bohr complementarity one must introduce wave particleduality in the discussion. This means one must consider both wave and particle behavior of light on an equal footing. Wave particleduality implies that one must A use the unitary evolution of the wave before the observation and B consider the particle aspect after the detection this is called the Heisenberg von Neumann ... recording of several photons. The above treatment formalizes wave particleduality for the double slit experiment. See also Afshar experiment Wave particleduality Quantum entanglement Quantum ...The Englert Greenberger duality relation relates the visibility, math V math , of interference fringes with the definiteness, or distinguishability, math D math , of the photons paths in quantum optics . ref name jagershimonyvaidman95 Gregg Jaeger , Abner Shimony , Lev Vaidman , Two interferometric complementarities , Phys. Rev. A, Vol. 51, 54 1995 ref ref name englert96 Berthold Georg Englert , Fringe ... name greenberger Daniel M. Greenberger , Allaine Yasin , Simultaneous wave and particle knowledge ... A P B , , math where math P A math and math P B math are the probabilities of finding that the particle ... p y h lambda cdot sin alpha math is the momentum of the particle along the y direction, math phi text ... B 2 . math And hence we get, for a single photon in a pure quantum state, the duality relation math ... s uncertainty principle itself. See also The Duality in Matter and Light Scientific ... Institute of experimental physics, Austria, http arxiv.org abs quant ph 0508091 for the details on the duality ... Englert Greenberger Duality Relation Category Quantum optics ... more details
refimprove date September 2011 The FRW CFT duality ref cite journal last1 Yasuhiro first1 Sekino authorlink1 Yasuhiro Sekino last2 Susskind first2 Leonard authorlink2 Leonard Susskind date 28 Oct. 2009 title Census Taking in the Hat FRW CFT Duality. journal Phys. Rev. volume 80 issue 8 series D pages 083531 doi 10.1103 PhysRevD.80.083531 url http prd.aps.org abstract PRD v80 i8 e083531 archiveurl http arxiv.org abs arXiv 0908.3844 archivedate 5 Apr 2010 ref ref cite journal last1 Susskind first1 Leonard authorlink1 Leonard Susskind title The Census taker s hat date 5 Oct 2007 url http arxiv.org abs arXiv 0710.1129 ref ref cite journal last1 Bousso first1 Raphael authorlink1 Raphael Bousso last2 Susskind first2 Leonard authorlink2 Leonard Susskind date 22 Jul 2011 title The Multiverse Interpretation of Quantum Mechanics url http arxiv.org abs arXiv 1105.3796 ref is a conjectured duality for Friedmann Robertson Walker model s inspired by the AdS CFT correspondence . It assumes that the cosmological constant is exactly zero, which is only the case for models with exact unbroken supersymmetry . Because the energy density doesn t approach zero as we approach spatial infinity, the metric isn t asymptotically flat . This isn t an asymptotically cold solution. In eternal inflation , our universe passes through a series of phase transitions with progressively lower cosmological constant. Our current phase has a cosmological constant of size math 10 123 math , which is conjectured to be metastable in string theory . It is possible our universe might tunnel into a supersymmetric phase with an exactly zero cosmological constant. In fact, any particle in eternal inflation will eventually terminate in a phase with exactly zero or negative cosmological constant. The phases with negative cosmological constant will end in a Big Crunch . Shenkar and Leonard Susskind called this the Census Taker s Hat . The conformal compactification of the terminal phase has a Penrose diagram shaped like ... more details
In mathematics, Fenchel s duality theorem is a result in the theory of convex functions named after Werner Fenchel . Let &fnof be a proper convex function on R sup n sup and let g be a proper concave function on R sup n sup . Then, if regularity conditions are satisfied, math min x f x g x max p g star p f star p . , math where &fnof sup sup is the convex conjugate of &fnof also referred to as the Fenchel&ndash Legendre transform and g sub sub is the concave conjugate of g . That is, math f star left x right sup left left. left langle x , x right rangle f left x right right x in mathbb R n right math math g star left x right inf left left. left langle x , x right rangle g left x right right x in mathbb R n right math Mathematical theorem Let X and Y be Banach spaces , math f X to mathbb R cup infty math and math g Y to mathbb R cup infty math be convex functions and math A X to Y math be a bounded operator bounded linear map . Then the Fenchel problems math p inf x in X f x g Ax math math d sup y in Y f A y g y math satisfy weak duality , i.e. math p geq d math . Note that math f ,g math are the convex conjugates of f , g respectively, and math A math is the adjoint operator . The perturbation function for this dual problem is given by math F x,y f x g Ax y math . Suppose that f , g , and A satisfy either f and g are lower semi continuous and math 0 in operatorname core operatorname dom g A operatorname dom f math where math operatorname core math is the algebraic interior and math operatorname dom h math where h is some function is the set math z h z infty math , or math A operatorname dom f cap operatorname cont g neq emptyset math where math operatorname cont math are the points where the function is continuous function continuous . Then strong duality holds, i.e. math p d math . If math d in mathbb R math then supremum is attained. ref cite book title Techniques of Variational Analysis last1 Borwein first1 Jonathan last2 Zhu first2 Qiji year 2005 publisher ... more details
Quantum mechanics cTopic Fundamental concepts Wave particleduality postulates that all particle s exhibit both wave and Subatomic particleparticle properties. A central concept of quantum mechanics , this duality ... , in which wave particleduality is one aspect of the concept of complementarity physics complementarity ... the quintessential example of wave particleduality. Electromagnetic radiation propagates following ... construct to explain the observed wave particleduality. In this view, each particle has a well defined ... Encyclopedia of Philosophy. ref , the wave particleduality is not a property of matter itself, but an appearance ... emphasizing the action of gravity in relation to wave particleduality were conducted in the 1970s ... A. Zeilinger year 1999 month 14 October title Wave particleduality of C sub 60 sub journal Nature ... on vibrating surface as a model of wave particleduality localized droplet creates periodical ... has to return to the initial state . Treatment in modern quantum mechanics Wave particleduality is deeply ... of current thinking on the phenomena historically called wave particleduality . See also .... Visualization Below is an illustration of how wave particleduality is consistent with De Broglie ... image1 Wave particleduality p known.svg caption1 Wave particle with a measurable wavelength has ... in position x or time t are both large. width2 250 image2 Wave particleduality ... some confined region of space. Alternative views Wave particleduality is an ongoing conundrum in modern physics. Most physicists accept wave particleduality as the best explanation for a broad ... proposes that there is no duality, but rather a system exhibits both particle properties and wave properties ... Afshar s ref Afshar S.S. et al Paradox in Wave ParticleDuality. Found. Phys. 37, 295 2007 http ..., in his search for a Unified Field Theory , did not accept wave particleduality, wrote ref Paul Arthur ... approach to wave particleduality Relational quantum mechanics is developed which regards the detection ... more details
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