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
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 disambiguation needed date January 2012 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 www.apple.com hotnews articles 2001 06 duality accessdate March 8, 2011 archiveurl http web.archive.org web 20011217112524 http www.apple.com hotnews articles 2001 06 duality archivedate December 17, 2001 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 ... http www.digitalproducer.com aHTM 2001 feats duality duality.htm accessdate March 8, 2011 ref Post ... release had all new music by Alan Derian. Reception Reaction to Duality was mostly positive, generating ... links imdb title id 1741653 title Duality youtube E03vPU Hsiw Duality http www.ifilm.com ifilmdetail 1011883 Duality on ifilm Non canon Star Wars DEFAULTSORT Duality Star Wars Fan Film Category Star ... 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
uses see Mathematics disambiguation and Math disambiguation . File Euclid.jpg thumb Euclid , Greek ... . ref Mathematics from Greek language Greek m th ma , knowledge, study, learning is the study ... reasoning often provides insight or predictions. Through the use of abstraction mathematics abstraction and logic al reasoning , mathematics developed from counting , calculation , measurement ... mathematics has been a human activity for as far back as History of Mathematics written records exist. Logic Rigorous arguments first appeared in Greek mathematics , most notably in Euclid Euclid s Euclid s Elements Elements . Mathematics developed at a relatively slow pace until the Renaissance , when ... of Mathematics 1. Newton and Leibniz , BBC Radio 4 , 27 09 2010. ref Carl Friedrich Gauss 1777 1855 referred to mathematics as the Queen of the Sciences . ref name Waltershausen Waltershausen ref Benjamin Peirce 1809 1880 called mathematics the science that draws necessary conclusions . ref Peirce, p. 97. ref David Hilbert said of mathematics We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual ... , Basel, Birkh user 1992 . ref Albert Einstein 1879 1955 stated that as far as the laws of mathematics ... . ref name certain Mathematics is used throughout the world as an essential tool in many fields, including natural science , engineering , medicine , and the social sciences . Applied mathematics , the branch of mathematics concerned with application of mathematical knowledge to other fields ... in pure mathematics , or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered. ref Peterson ref Etymology The word mathematics comes from ... of which mean to learn . The word mathematics in Greek came to have the narrower and more technical ... 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 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 mathematics, Alvis Curtis duality is a Dualitymathematicsduality 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 ... books?id LvvuAAAAMAAJ publisher John Wiley & Sons location New York series Pure and Applied Mathematics ... 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
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 ... 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 1990 , eds. A. Carboni, M. C. Pedicchio and G. Rosolini, Lectures Notes in Mathematics 1488 , Springer ... Category Harmonic analysis Category Topological groups Category Duality theories ... more details
In Galois cohomology , local Tate duality or simply local duality is a Dualitymathematicsduality 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 dualitymathematics 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 ... 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 ... systems publisher Princeton University Press year 2000 series Hermann Weyl Lectures, Annals of Mathematics ..., New York series Springer Monographs in Mathematics isbn 978 3 540 42192 4 mr 1867431 year 2002 , translation ... 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
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 the mathematics mathematical area of order theory , every partially ordered set P gives rise to a dual or opposite partially ordered set which is often denoted by P sup op sup or P sup d sup . This dual order P sup op sup is defined to be the set with the inverse order , i.e. x y holds in P sup op sup if and only if y x holds in P . It is easy to see that this construction, which can be depicted by flipping the Hasse diagram for P upside down, will indeed yield a partially ordered set. In a broader sense, two poset s are also said to be duals if they are dually isomorphic , i.e. if one poset is order isomorphism order isomorphic to the dual of the other. The importance of this simple definition stems from the fact that each and every definition and theorem of order theory can readily be transferred to the dual order. Formally, this is captured by the Duality Principle for ordered sets If a given statement is valid for all partially ordered sets, then its dual statement, obtained by inverting the direction of all order relations and by dualizing all order theoretic definitions involved, is also valid for all partially ordered sets. If a statement or definition is equivalent to its dual then it is said to be self dual . Note that the consideration of dual orders is so fundamental that it often occurs implicitly when writing for the dual order of without giving any prior definition of this new symbol. Examples Naturally, there are a great number of examples for concepts that are dual Greatest element Greatest elements and least elements Maximal element Maximal elements and minimal elements Least upper bound s suprema, math vee math and greatest lower bound s infima, math wedge math Upper set Upper sets and lower sets ideal order theory Ideals and filter mathematics filters Closure operator s and kernel operator s. Examples of notions which are self dual include Being a complete ... Cambridge University Press isbn 978 0 521 78451 1 year 2002 Category Order theory Category Duality ... more details
s homogeneous coordinates are represented as a vector in 3x1 matrix mathematics matrix form, then the duality ... 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 ... on a line form a projective range , a concept dual to a pencil mathematics pencil of lines on a point ... 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 ... a projective correlation . A correlation of order two an Involution mathematics involution is called .... 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 ... more details
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 Visibility and Which Way Information An Inequality , Phys. Rev. Lett., Vol. 77, 2154 1996 ref ref name greenberger Daniel M. Greenberger , Allaine Yasin , Simultaneous wave and particle knowledge in a neutron interferometer , Phys. Lett. A 128, 391, 1988 ref As an inequality math D 2 V 2 le 1 , math The relationship was first experimentally shown by Greenberger and Yassin in 1988. It was later ... a year later it was also mentioned by Englert, in 1996. The mathematics of two slit diffraction Image ... 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 particle duality. Of this experiment, Richard Feynman once said that it has in it the heart of quantum ... B 2 . math And hence we get, for a single photon in a pure quantum state, the duality relation math ... formulation of Bohr complementarity one must introduce wave particle duality in the discussion ... particle duality implies that one must A use the unitary evolution of the wave before the observation ... recording of several photons. The above treatment formalizes wave particle duality for the double slit experiment. See also Afshar experiment Wave particle duality Quantum entanglement Quantum ... 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
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
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
Quantum mechanics cTopic Fundamental concepts Wave particle duality postulates that all particle s exhibit both wave and Subatomic particle particle properties. A central concept of quantum mechanics , this duality ... as a fundamental property of the Universe, while alternative interpretations explain the duality ... , in which wave particle duality is one aspect of the concept of complementarity physics complementarity ... of theory The idea of duality originated in a debate over the nature of light and matter that dates ... the quintessential example of wave particle duality. Electromagnetic radiation propagates following ... construct to explain the observed wave particle duality. In this view, each particle has a well defined ... Encyclopedia of Philosophy. ref , the wave particle duality is not a property of matter itself, but an appearance ... emphasizing the action of gravity in relation to wave particle duality were conducted in the 1970s ... A. Zeilinger year 1999 month 14 October title Wave particle duality of C sub 60 sub journal Nature ... on vibrating surface as a model of wave particle duality localized droplet creates periodical ... has to return to the initial state . Treatment in modern quantum mechanics Wave particle duality is deeply ... discuss it as such. In the formalism mathematics formalism of the theory, all the information about ... of current thinking on the phenomena historically called wave particle duality . See also .... Visualization Below is an illustration of how wave particle duality is consistent with De Broglie ... image1 Wave particle duality p known.svg caption1 Wave particle with a measurable wavelength has ... in position x or time t are both large. width2 250 image2 Wave particle duality ... some confined region of space. Alternative views Wave particle duality is an ongoing conundrum in modern physics. Most physicists accept wave particle duality as the best explanation for a broad ... proposes that there is no duality, but rather a system exhibits both particle properties and wave properties ... 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
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
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
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