# 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 $\mathbf R ^ {+}$, having certain characteristic properties, conditioned by properties of the set $\Omega$ 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 $\Omega \rightarrow \mathbf R ^ {+}$ turns out to be a morphism of some structure in $\Omega$ onto one of the (algebraic) structures in $\mathbf R ^ {+}$, 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: $\alpha$)– $\epsilon$)). In more abstract situations it is natural to use an ordered semi-ring instead of $\mathbf R ^ {+}$( 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 $x$ of a semi-ordered space $P$ is the number

$$| x | = x ^ {+} + x ^ {-} ,$$

where $x ^ {+}$( $x ^ {-}$) is the positive (negative) part of $x$. Here, as for real numbers,

$\alpha$) $| x | \geq x , - x$; $| x | = | - x |$;

$\beta$) $| x | = 0 \iff x = 0$( 0 is the zero in $P$).

2) The modulus of an element $x$ of a separable pre-Hilbert space $H$, in particular, a finite-dimensional vector space, is the number

$$| x | = \langle x , x \rangle ^ {1/2} ,$$

where $\langle \cdot , \cdot \rangle$ is the inner product in $H$. This is a norm in $H$ and thus

$\gamma$) $| x + y | \leq | x | + | y |$;

$\delta$) $| \lambda x | = | \lambda | | x |$, $\lambda$ a scalar.

3) The modulus of an element $x$ of a locally compact skew-field is the number

$$| x | = \ \frac{\mu ( x S ) }{\mu ( S) } \ ( x \neq 0 ) \ \ \textrm{ or } \ \ 0 ( x = 0 ) ,$$

where $\mu$ is a Haar measure on the additive group of $K$ and $S$ is a measurable subset. Here, as for numbers from $\mathbf R$, $\mathbf C$, $\mathbf H$,

$\epsilon$) $| x y | = | x | | y |$.

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

4) The modulus of an endomorphism $A$ of a vector space $V$ over a field $K$( a special case is the modulus of an automorphism) is the number $\mathop{\rm mod} _ {V} ( A)$, which turns out to be simply equal to $\mathop{\rm mod} _ {K} ( \mathop{\rm det} A ) = | \mathop{\rm det} A |$, where $| \cdot |$ is the modulus of Example 3).