What is "duality"

Duality \Du"al"i*ty\, n. [L. dualitas: cf. F. dualit['e].] The quality or condition of being two or twofold; dual character or usage.

late 14c., from Old French dualité (14c.), from Late Latin dualitas, from Latin dualis (see dual).

n. 1 A classification into two subclasses or opposed parts. 2 (context projective geometry English) The interchangeability of points and planes. 3 (context mathematics physics English) The mathematical equivalence of two seemingly different theoretical descriptions of a physical system.

1. n. being twofold; a classification into two opposed parts or subclasses; "the dichotomy between eastern and western culture" [syn: dichotomy]

2. (physics) the property of matter and electromagnetic radiation that is characterized by the fact that some properties can be explained best by wave theory and others by particle theory [syn: wave-particle duality]

3. (geometry) the interchangeability of the roles of points and planes in the theorems of projective geometry

Duality may refer to:

Duality is a collaborative album by Lisa Gerrard and Pieter Bourke released in 1998. It was Lisa Gerrard's second post- Dead Can Dance album after The Mirror Pool from 1995. Lisa Gerrard and Brendan Perry released their last album in 1996 and had officially disbanded Dead Can Dance earlier in 1998.

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 fields. 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 (E) is the dual of the magnetizing field (H).
• The electric displacement field (D) is the dual of the magnetic flux density (B).
• Faraday's law of induction is the dual of Ampère's circuital 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.
• 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 magnetic motor;
• Electrets are the dual of 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 hypothetical magnetic monopole is the dual of electric charge.

In geometry a striking feature of projective planes is the symmetry of the roles played by points and lines in the definitions and theorems, and ( plane) duality is the formalization of this concept. There are two approaches to the subject of duality, one through language and the other a more functional approach through special mappings. These are completely equivalent and either treatment has as its starting point the axiomatic version of the geometries under consideration. In the functional approach there is a map between related geometries that is called a duality. Such a map can be constructed in many ways. The concept of plane duality readily extends to space duality and beyond that to duality in any finite-dimensional projective geometry.

In mathematical optimization theory, duality or the duality principle is the principle that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem. The solution to the dual problem provides a lower bound to the solution of the primal (minimization) problem. However in general the optimal values of the primal and dual problems need not be equal. Their difference is called the duality gap. For convex optimization problems, the duality gap is zero under a constraint qualification condition.

"Duality" is a song by the American heavy metal band Slipknot, the song was released as the first single from their third album Vol. 3: (The Subliminal Verses), released in 2004. The music video was listed as Roadrunner Records' greatest video of all-time.

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.

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 — deformation
• stress — strain
• stiffness method — flexibility method

Duality (stylized as Du∆lity) is the debut mixtape from Captain Murphy, the animated rapper alter-ego of experimental musician Flying Lotus. It was initially released as a 34-minute short film featuring songs performed by Murphy as well as archived cult footage (including footage involving Heaven's Gate co-founder Marshall Applewhite), pornography, cartoons and 1980s films. A deluxe edition was released for download on November 28, 2012 with tracks separated, as well as bonus tracks and instrumentals.

In the mathematical area of order theory, every partially ordered set P gives rise to a dual (or opposite) partially ordered set which is often denoted by P or P. This dual order P is defined to be the set with the inverse order, i.e. x ≤ y holds in P 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 posets are also said to be duals if they are dually isomorphic, i.e. if one poset is order isomorphic to the dual of the other.

The importance of this simple definition stems from the fact that every definition and theorem of order theory can readily be transferred to the dual order. Formally, this is captured by the Duality Principle for ordered sets:

If a 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.

In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A. Such involutions sometimes have fixed points, so that the dual of A is A itself. For example, Desargues' theorem is self-dual in this sense under the standard duality in projective geometry.

In mathematical contexts, duality has numerous meanings although it is "a very pervasive and important concept in (modern) mathematics" and "an important general theme that has manifestations in almost every area of mathematics".

Many mathematical dualities between objects of two types correspond to pairings, bilinear functions from an object of one type and another object of the second type to some 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 distributions and the associated test functions corresponds to the pairing in which one integrates a distribution against a test function, and Poincaré duality corresponds similarly to intersection number, viewed as a pairing between submanifolds of a given manifold.

From a category theory viewpoint, duality can also be seen as a functor, at least in the realm of vector spaces. This functor assigns to each space its dual space, and the pullback construction assigns to each arrow its dual .

Duality is a 2005 album from the 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", a song that was inspired by the Kevin Smith film Dogma. The tenth track "Got Me Going" was played on the radio in the background of the movie Hot Rod. Janel Elizabeth has a cover of the song "Swimming Upstream" streamed on her Myspace.

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. Duality also reached #137 on the Billboard 200, and also #2 for 18 weeks on Heatseekers Albums.

In the 2005 Melrock Awards, Duality was named the 3rd best modern rock album of 2005. The song "Fallen Angels" was named 7th best modern rock song, and "Every Little Thing She Does Is Magic" was named 5th best cover version.

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 of electricity and magnetism.

Here is a partial list of electrical dualities:

• voltage – current
• parallel – serial (circuits)
• resistance – conductance
• impedance – admittance
• capacitance – inductance
• reactance – susceptance
• short circuit – open circuit
• Kirchhoff's current law – Kirchhoff's voltage law.
• Thévenin's theorem – Norton's theorem

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 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' .

Some compare the concept of a duality to that of Yin and Yang, i.e. two mutually defining opposites.

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. identifies four dualities that exist in Communities of Practice: participation-reification, designed-emergent, identification-negotiability and local-global.