Namespaces
Variants
Actions

Difference between revisions of "Modulus"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
A numerical characteristic of various mathematical objects. Usually the value of a modulus is a non-negative real number, an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m0645201.png" />, having certain characteristic properties, conditioned by properties of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m0645202.png" /> of objects under discussion. The notion of a modulus figures in various branches of mathematics, although sometimes under other names — [[Absolute value|absolute value]]; [[Norm|norm]], etc. All of them, in essence, are generalizations of the idea of the [[Absolute value|absolute value]] of a real or complex number (but the term modulus usually means a generalization of special form). Here the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m0645203.png" /> turns out to be a morphism of some structure in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m0645204.png" /> onto one of the (algebraic) structures in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m0645205.png" />, 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: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m0645206.png" />)–<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m0645207.png" />)). In more abstract situations it is natural to use an ordered semi-ring instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m0645208.png" /> (this conception of a modulus is satisfied by, for example, a [[Measure|measure]], a [[Capacity|capacity]], a [[Mass|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|modulus of an annulus]], the [[Moduli of a Riemann surface|moduli of a Riemann surface]], and the modulus of continuity or smoothness (cf. [[Continuity, modulus of|Continuity, modulus of]]; [[Smoothness, 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|extremal length]] instead of the modulus).
+
<!--
 +
m0645201.png
 +
$#A+1 = 47 n = 0
 +
$#C+1 = 47 : ~/encyclopedia/old_files/data/M064/M.0604520 Modulus
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
Examples. 1) The modulus of an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m0645209.png" /> of a [[Semi-ordered space|semi-ordered space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m06452010.png" /> is the number
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m06452011.png" /></td> </tr></table>
+
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|absolute value]]; [[Norm|norm]], etc. All of them, in essence, are generalizations of the idea of the [[Absolute value|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|measure]], a [[Capacity|capacity]], a [[Mass|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|modulus of an annulus]], the [[Moduli of a Riemann surface|moduli of a Riemann surface]], and the modulus of continuity or smoothness (cf. [[Continuity, modulus of|Continuity, modulus of]]; [[Smoothness, 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|extremal length]] instead of the modulus).
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m06452012.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m06452013.png" />) is the positive (negative) part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m06452014.png" />. Here, as for real numbers,
+
Examples. 1) The modulus of an element  $  x $
 +
of a [[Semi-ordered space|semi-ordered space]]  $  P $
 +
is the number
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m06452015.png" />) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m06452016.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m06452017.png" />;
+
$$
 +
| x |  = x  ^ {+} + x  ^ {-} ,
 +
$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m06452018.png" />) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m06452019.png" /> (0 is the zero in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m06452020.png" />).
+
where  $  x  ^ {+} $(
 +
$  x  ^ {-} $)  
 +
is the positive (negative) part of  $  x $.  
 +
Here, as for real numbers,
  
2) The modulus of an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m06452021.png" /> of a separable [[Pre-Hilbert space|pre-Hilbert space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m06452022.png" />, in particular, a finite-dimensional vector space, is the number
+
$  \alpha $)  
 +
$  | x | \geq  x , - x $;
 +
$  | x | = | - x | $;
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m06452023.png" /></td> </tr></table>
+
$  \beta $)
 +
$  | x | = 0 \iff x = 0 $(
 +
0 is the zero in  $  P $).
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m06452024.png" /> is the [[Inner product|inner product]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m06452025.png" />. This is a norm in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m06452026.png" /> and thus
+
2) The modulus of an element  $  x $
 +
of a separable [[Pre-Hilbert space|pre-Hilbert space]] $  H $,
 +
in particular, a finite-dimensional vector space, is the number
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m06452027.png" />) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m06452028.png" />;
+
$$
 +
| x |  = \langle  x , x \rangle  ^ {1/2} ,
 +
$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m06452029.png" />) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m06452030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m06452031.png" /> a scalar.
+
where  $  \langle  \cdot , \cdot \rangle $
 +
is the [[Inner product|inner product]] in  $  H $.  
 +
This is a norm in  $  H $
 +
and thus
  
3) The modulus of an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m06452032.png" /> of a [[Locally compact skew-field|locally compact skew-field]] is the number
+
$  \gamma $)  
 +
| x + y | \leq  | x | + | y | $;
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m06452033.png" /></td> </tr></table>
+
$  \delta $)
 +
$  | \lambda x | = | \lambda | | x | $,
 +
$  \lambda $
 +
a scalar.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m06452034.png" /> is a [[Haar measure|Haar measure]] on the additive group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m06452035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m06452036.png" /> is a measurable subset. Here, as for numbers from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m06452037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m06452038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m06452039.png" />,
+
3) The modulus of an element  $  x $
 +
of a [[Locally compact skew-field|locally compact skew-field]] is the number
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m06452040.png" />) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m06452041.png" />.
+
$$
 +
| x |  = \
  
A generalization of this idea is the [[Modulus of an automorphism|modulus of an automorphism]].
+
\frac{\mu ( x S ) }{\mu ( S) }
 +
\
 +
( x \neq 0 ) \ \
 +
\textrm{ or } \ \
 +
0  ( x = 0 ) ,
 +
$$
 +
 
 +
where  $  \mu $
 +
is a [[Haar measure|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 $,
  
4) The modulus of an endomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m06452042.png" /> of a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m06452043.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m06452044.png" /> (a special case is the modulus of an automorphism) is the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m06452045.png" />, which turns out to be simply equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m06452046.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064520/m06452047.png" /> is the modulus of Example 3).
+
$  \epsilon $)  
 +
$  | x y | = | x | | y | $.
  
 +
A generalization of this idea is the [[Modulus of an automorphism|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).
  
 
====Comments====
 
====Comments====
 
Often a modulus is simply some numerical parameter on which the mathematical object under consideration depends. For instance, the [[Modulus of an elliptic integral|modulus of an elliptic integral]], the complementary modulus (in [[Jacobi elliptic functions|Jacobi elliptic functions]]) or the modulus of a [[Congruence|congruence]]. Cf. also [[Norm on a field|Norm on a field]]; [[Valuation|Valuation]].
 
Often a modulus is simply some numerical parameter on which the mathematical object under consideration depends. For instance, the [[Modulus of an elliptic integral|modulus of an elliptic integral]], the complementary modulus (in [[Jacobi elliptic functions|Jacobi elliptic functions]]) or the modulus of a [[Congruence|congruence]]. Cf. also [[Norm on a field|Norm on a field]]; [[Valuation|Valuation]].

Latest revision as of 08:01, 6 June 2020


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).

Comments

Often a modulus is simply some numerical parameter on which the mathematical object under consideration depends. For instance, the modulus of an elliptic integral, the complementary modulus (in Jacobi elliptic functions) or the modulus of a congruence. Cf. also Norm on a field; Valuation.

How to Cite This Entry:
Modulus. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Modulus&oldid=16003
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article