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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.
The max-flow min-cut theorem is a special case of the strong duality theorem: flow-maximization is the primal LP, and cut-minimization is the dual LP. See Max-flow min-cut theorem#Linear program formulation. Other graph-related theorems can be proved using the strong duality theorem, in particular, Konig's theorem. [9]
The duality gap is zero if and only if strong duality holds. Otherwise the gap is strictly positive and weak duality holds. [5] In computational optimization, another "duality gap" is often reported, which is the difference in value between any dual solution and the value of a feasible but suboptimal iterate for the primal problem.
For example, the dual of (A & B ∨ C) would be (¬A ∨ ¬B & ¬C). The dual of a formula φ is notated as φ*. The Duality Principle states that in classical propositional logic, any sentence is equivalent to the negation of its dual. [4] [7] Duality Principle: For all φ, we have that φ = ¬(φ*). [4] [7] Proof: By induction on complexity ...
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.
A duality that respects the orderings in question is known as a Galois connection. An example is the standard duality in Galois theory mentioned in the introduction: a bigger field extension corresponds—under the mapping that assigns to any extension L ⊃ K (inside some fixed bigger field Ω) the Galois group Gal (Ω / L) —to a smaller ...
Download as PDF; Printable version; ... The Duality Theory and the Basic Isomorphisms of Group Systems and Nets and Co-Nets of Group Systems. ... Duality Theorems. In ...
In mathematics, a dual system, dual pair or a duality over a field is a triple (,,) consisting of two vector spaces, and , over and a non-degenerate bilinear map:. In mathematics , duality is the study of dual systems and is important in functional analysis .