# duality theorem

## duality theorem

[dü′al·əd·ē ‚thir·əm]
(mathematics)
A theorem which asserts that for a given n-dimensional space, the (n-p) dimensional homology group is isomorphic to a p-dimensional cohomology group for each p = 0, …, n, provided certain conditions are met.
Let G be either a compact group or a discrete group, let X be its character group, and let G′ be the character group of X ; then there is an isomorphism of G onto G′ so that the groups G and G′ may be identified.
If either of two dual linear-programming problems has a solution, then so does the other.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
References in periodicals archive ?
Our idea is that we first show the metric D is identical with the first Wasserstein metric on P(K) thanks to the duality theorem of Kantorovich-Rubinstein [KR58] (see also Villani's book [V09, Particular Case 5.16]), and then use several definitions for weak contractions which are equivalent to Browder's definition.
Drawing inspiration from Brouwer's work and Poincare's results, in 1922 Alexander formulated his own duality theorem, attaining to the contextual properties of mathematical objects through the analysis of their internal structure (Lautman, 2011).
Furthermore, if the weak duality Theorem 20 holds for all feasible solutions of the problems (MP) and (MDI), then ([bar.x], [bar.[lambda]], [bar.[mu]]) is a weakly efficient solution of (MDI).
Terai Alexander duality theorem and Stanley-Reisner rings Surikaisekikenkyusho Kokyu- ruko(1999) no.
Because [Q.sub.1] [??] 0, the convex set S has an interior point and objective function [[xi].sup.T] [Q.sub.1][xi] + [([q.sub.1] + x).sup.T] [xi] + [r.sub.1] in the left-hand side of (10) is bounded below on S, and from the conic duality theorem, we know that (10) can be transformed into the following problem:
Both model (5) and its dual problem are feasible, so by the duality theorem of linear programming they have bounded optimal solutions.
By the weak duality theorem, ((u,v), [bar.[tau]], [bar.[lambda]], [bar.v]) is an efficient solution of (FD2).
Invoking the Duality Theorem of Linear Programming , we arrive at
Chapters discuss duality, linear mappings, matrices, determinant and trace, spectral theory, Euclidean structure, calculus of vector- and matrix-valued functions, matrix inequalities, kinematics and dynamics, convexity, the duality theorem, normed liner spaces, linear mappings between normed linear spaces, positive matrices, and solutions of systems of linear equations.
Keywords: Evolution equation, Gross Laplacian, potential function, white noise analysis, generalized functions, convolution operator, Laplace transform, duality theorem.
Teo: A converse duality theorem on higher-order dual models in nondifferentiable mathematical programming, Optim.
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