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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.
For example, Desargues' theorem is self-dual in this sense under the standard duality in projective geometry. In mathematical contexts, duality has numerous meanings. [1] It has been described as "a very pervasive and important concept in (modern) mathematics" [2] and "an important general theme that has manifestations in almost every area of ...
But for some classes of functions, it is possible to get an explicit formula for g(). Solving the primal and dual programs together is often easier than solving only one of them. Examples are linear programming and quadratic programming. A better and more general approach to duality is provided by Fenchel's duality theorem. [18]: Sub.3.3.1
The strong duality theorem says that if one of the two problems has an optimal solution, so does the other one and that the bounds given by the weak duality theorem are tight, i.e.: max x c T x = min y b T y. The strong duality theorem is harder to prove; the proofs usually use the weak duality theorem as a sub-routine.
In projective geometry, duality or plane duality is a formalization of the striking symmetry of the roles played by points and lines in the definitions and theorems of projective planes. There are two approaches to the subject of duality, one through language (§ Principle of duality) and the other a more functional approach through special ...
The idea of mathematical duality was first noticed as projective duality. There it appears as the idea of interchanging dimension k and codimension k+1 in propositions of projective geometry. A large number of duality theories have now been created in mathematics, ranging as far as optimization theory and theoretical physics.
In mathematics, Alexander duality refers to a duality theory initiated by a result of J. W. Alexander in 1915, and subsequently further developed, particularly by Pavel Alexandrov and Lev Pontryagin. It applies to the homology theory properties of the complement of a subspace X in Euclidean space , a sphere , or other manifold .
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 – series (circuits) resistance – conductance; voltage division – current division ...