dual

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dual

Maths Logic (of structures or expressions) having the property that the interchange of certain pairs of terms, and usually the distribution of negation, yields equivalent structures or expressions
Collins Discovery Encyclopedia, 1st edition © HarperCollins Publishers 2005

dual

i. Duplicated controls permitting either an instructor and a pupil or the pilot and the copilot to operate controls.
ii. Instructional flying hours or sorties logged in a flying log by the pupil.
An Illustrated Dictionary of Aviation Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved

dual

(mathematics)
Every field of mathematics has a different meaning of dual. Loosely, where there is some binary symmetry of a theory, the image of what you look at normally under this symmetry is referred to as the dual of your normal things.

In linear algebra for example, for any vector space V, over a field, F, the vector space of linear maps from V to F is known as the dual of V. It can be shown that if V is finite-dimensional, V and its dual are isomorphic (though no isomorphism between them is any more natural than any other).

There is a natural embedding of any vector space in the dual of its dual:

V -> V'': v -> (V': w -> wv : F)

(x' is normally written as x with a horizontal bar above it). I.e. v'' is the linear map, from V' to F, which maps any w to the scalar obtained by applying w to v. In short, this double-dual mapping simply exchanges the roles of function and argument.

It is conventional, when talking about vectors in V, to refer to the members of V' as covectors.
This article is provided by FOLDOC - Free Online Dictionary of Computing (foldoc.org)
References in periodicals archive ?
Bounds on the deviation from perfect reconstruction are obtained for approximately dual frames constructed via perturbation theory.
Given a frame {[f.sub.k]} for H, it is well known that there exists at least one dual frame {[g.sub.k]}, that is, a frame for which
SHC seeks details of PTI leader Faisal Vawda's dual nationality
During an earlier hearing, the apex court asked if the law prevents members of the armed forces from holding dual nationalities.
He further stated that they received data of more than 36,000 officers wherein 655 government officials confessed of holding dual nationality and 152 concealed.
Furthermore, 665 officers are married to women who possess dual nationality and five to women who are foreign nationals.
The apex court has issued notices to the four senators to clarify their positions after the ECP said some of them had already submitted affidavits to the commission giving up their dual nationality.
According to a report submitted to the Supreme Court, the number of civil servants holding dual nationalities is 64 in Punjab, five in Sindh, nine in Khyber Pakhtunkhwa (KP), eight in Balochistan, 28 in Azad Kashmir, and one in Gilgit Baltistan.
Andrews [11] determined that students who had taken dual credit courses had higher college GPAs than did students who had not taken dual credit courses.
[12] stated that the greatest benefit of successful dual credit course completion was a higher GPA for students when they matriculate into college.
Segovia-Vargas, "Dual band high efficiency class ce power amplifier based on CRLH diplexer," Progress In Electromagnetics Research, Vol.
Scucchia, "A new structure for the design of dual band power amplifiers," 2011 Workshop on Integrated Nonlinear Microwave and Millimetre-wave Circuits (INMMIC), 1-4, 2011.