# Modulus

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A numerical characteristic of various mathematical objects. Usually the value of a modulus is a non-negative real number, an element of , having certain characteristic properties, conditioned by properties of the set of objects under discussion. The notion of a modulus figures in various branches of mathematics, although sometimes under other names — absolute value; norm, etc. All of them, in essence, are generalizations of the idea of the absolute value of a real or complex number (but the term modulus usually means a generalization of special form). Here the function turns out to be a morphism of some structure in onto one of the (algebraic) structures in , among which the most important ones are the order, the addition and the multiplication. In this connection the basic properties of the absolute value must be preserved (see below: )– )). In more abstract situations it is natural to use an ordered semi-ring instead of (this conception of a modulus is satisfied by, for example, a measure, a capacity, a mass, etc.). Finally, the term modulus denotes numerical characteristics of other objects, such as, for example, the moduli of a plane domain, the modulus of an annulus, the moduli of a Riemann surface, and the modulus of continuity or smoothness (cf. Continuity, modulus of; Smoothness, modulus of) (and even moduli in the theory of elasticity (compression, shear)). However, in all these cases it is possible to introduce a value functionally depending on the modulus and more adequately reflecting the nature of the objects under discussion (for example, for a family of curves, the extremal length instead of the modulus).

Examples. 1) The modulus of an element of a semi-ordered space is the number where ( ) is the positive (negative) part of . Here, as for real numbers, ) ; ; ) (0 is the zero in ).

2) The modulus of an element of a separable pre-Hilbert space , in particular, a finite-dimensional vector space, is the number where is the inner product in . This is a norm in and thus ) ; ) , a scalar.

3) The modulus of an element of a locally compact skew-field is the number where is a Haar measure on the additive group of and is a measurable subset. Here, as for numbers from , , , ) .

A generalization of this idea is the modulus of an automorphism.

4) The modulus of an endomorphism of a vector space over a field (a special case is the modulus of an automorphism) is the number , which turns out to be simply equal to , where is the modulus of Example 3).