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

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.

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.
References in periodicals archive ?
Interestingly, the interim and current constitution of AJK does not place any restriction on Kashmiris with dual nationality contesting elections and holding membership of the Legislative Assembly.
The changing of the laws to allow dual citizenship will help address the concerns of those who are against the celebrations of Liberia at 172 and those in favor.
During today's hearing, the Federal Investigation Agency (FIA) submitted a list of around 1,000 dual nationals holding government offices.
The governments should give time to abandon citizenship to the employees who have acquired dual nationality during their service tenure and if they don't do so then proceedings should be initiated against them.
He added that government officers should not possess dual nationality and it would be better if the courts let the government and the Parliament take a decision on the matter.
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A three-judge bench headed by Chief Justice of Pakistan took up the suo moto case regarding dual national government officers.
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A three-member bench of the apex court headed by Chief Justice of Pakistan (CJP) Justice Mian Saqib Nisar heard the case related to the dual nationality of the judges and government officials.
In acknowledgment of this valuable educational opportunity, the Illinois Senate recently declared March 18 to 24 as "Dual Credit Week."
Considering the design difficulty, electrical performance, and tooling cost of dual polarized antenna unit, an optional implementation scheme is a dual polarized printed antenna.