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 ?
And we can prove that a, b and c are cyclic dualizing elements of Q.
Then there is a one-to-one correspondence between the set of cyclic dualizing elements in Q and the set of unary operations satisfying the condition CN in Theorem 2.
If 0 is a cyclic dualizing element of Q, then Q is strictly two-sided.
If Q is a two-sided Girard quantale, then the unique cyclic dualizing element is the least element 0.
Any complete lattice implication algebra is a Girard quantale with unique cyclic dualizing element 0.
According the above conclusions, we have a question : Whether the cyclic dualizing element must be the least element 0 if a Girard quantale has an unique cyclic dualizing element.
It is immediate to verify Q being a Girard quantale with the unique cyclic dualizing element e.