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''of a family of curves''
 
''of a family of curves''
  
 
A concept which, along with that of the modulus of a family of curves, is a general form of the definition of conformal invariants and lies at the basis of the method of the extremal metric (cf. [[Extremal metric, method of the|Extremal metric, method of the]]).
 
A concept which, along with that of the modulus of a family of curves, is a general form of the definition of conformal invariants and lies at the basis of the method of the extremal metric (cf. [[Extremal metric, method of the|Extremal metric, method of the]]).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e0371701.png" /> be a family of locally rectifiable curves on a [[Riemann surface|Riemann surface]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e0371702.png" />. The modulus problem is defined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e0371703.png" /> if there is a non-empty class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e0371704.png" /> of conformally-invariant metrics (cf. [[Conformally-invariant metric|Conformally-invariant metric]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e0371705.png" /> given on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e0371706.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e0371707.png" /> is square integrable in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e0371708.png" />-plane for every local uniformizing parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e0371709.png" /> and if
+
Let $  \Gamma $
 +
be a family of locally rectifiable curves on a [[Riemann surface|Riemann surface]] $  R $.  
 +
The modulus problem is defined for $  \Gamma $
 +
if there is a non-empty class $  P $
 +
of conformally-invariant metrics (cf. [[Conformally-invariant metric|Conformally-invariant metric]]) $  \rho ( z)  | d z | $
 +
given on $  R $
 +
such that $  \rho ( z) $
 +
is square integrable in the $  z $-
 +
plane for every local uniformizing parameter $  z ( = x + i y ) $
 +
and if
 +
 
 +
$$
 +
A _  \rho  ( R)  = {\int\limits \int\limits } _ { R } \rho  ^ {2} ( z)  d x  d y \ \
 +
\textrm{ and } \  L _  \rho  ( \Gamma )  =  \inf _ {\gamma \in \Gamma }
 +
\int\limits _  \gamma  \rho ( z)  | d z |
 +
$$
 +
 
 +
are not simultaneously equal to  $  0 $
 +
or  $  \infty $.  
 +
(Each of the above integrals is understood as a Lebesgue integral.) In this case the quantity
  
<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/e/e037/e037170/e03717010.png" /></td> </tr></table>
+
$$
 +
M ( \Gamma )  = \inf _ {\rho \in P } \
  
are not simultaneously equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717011.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717012.png" />. (Each of the above integrals is understood as a Lebesgue integral.) In this case the quantity
+
\frac{A _  \rho  ( R) }{[ L _  \rho  ( \Gamma ) ]  ^ {2} }
  
<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/e/e037/e037170/e03717013.png" /></td> </tr></table>
+
$$
  
is called the modulus of the family of curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717014.png" />. The reciprocal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717015.png" /> is called the extremal length of the family of curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717016.png" />.
+
is called the modulus of the family of curves $  \Gamma $.  
 +
The reciprocal of $  M ( \Gamma ) $
 +
is called the extremal length of the family of curves $  \Gamma $.
  
The modulus problem for a family of curves is often defined as follows: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717017.png" /> be the subclass of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717018.png" /> such that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717020.png" />,
+
The modulus problem for a family of curves is often defined as follows: Let $  P _ {L} $
 +
be the subclass of $  P $
 +
such that for $  \rho \in P _ {L} $
 +
and $  \gamma \in \Gamma $,
  
<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/e/e037/e037170/e03717021.png" /></td> </tr></table>
+
$$
 +
\int\limits _  \gamma  \rho ( z)  | d z |  \geq  1 .
 +
$$
  
If the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717022.png" /> is non-empty, then the quantity
+
If the set $  P _ {L} $
 +
is non-empty, then the quantity
  
<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/e/e037/e037170/e03717023.png" /></td> </tr></table>
+
$$
 +
M ( \Gamma )  = \inf _ {\rho \in P _ {L} }  A _  \rho  ( R)
 +
$$
  
is called the modulus of the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717024.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717025.png" /> is non-empty but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717026.png" /> is empty, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717027.png" /> is assigned the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717028.png" />. It is the latter definition of the modulus that is adopted below.
+
is called the modulus of the family $  \Gamma $.  
 +
If $  P $
 +
is non-empty but $  P _ {L} $
 +
is empty, then $  M ( \Gamma ) $
 +
is assigned the value $  \infty $.  
 +
It is the latter definition of the modulus that is adopted below.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717029.png" /> be a family of locally rectifiable curves on a Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717030.png" /> for which the modulus problem is defined, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717031.png" />. Then every metric from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717032.png" /> is an admissible metric for the modulus problem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717033.png" />. If in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717034.png" /> there is a metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717035.png" /> for which
+
Let $  \Gamma $
 +
be a family of locally rectifiable curves on a Riemann surface $  R $
 +
for which the modulus problem is defined, and let $  M ( \Gamma ) \neq \infty $.  
 +
Then every metric from $  P _ {L} $
 +
is an admissible metric for the modulus problem for $  \Gamma $.  
 +
If in $  P _ {L} $
 +
there is a metric $  \rho  ^ {*} ( z)  | dz | $
 +
for which
  
<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/e/e037/e037170/e03717036.png" /></td> </tr></table>
+
$$
 +
M ( \Gamma )  = \inf _ {\rho \in P _ {L} }  A _  \rho  ( R) ,
 +
$$
  
then this metric is called an extremal metric in the modulus problem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717037.png" />.
+
then this metric is called an extremal metric in the modulus problem for $  \Gamma $.
  
 
The fundamental property of the modulus is its conformal invariance.
 
The fundamental property of the modulus is its conformal invariance.
  
Theorem 1. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717039.png" /> be two conformally-equivalent Riemann surfaces, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717040.png" /> be a univalent conformal mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717041.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717042.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717043.png" /> be a family of locally rectifiable curves given on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717044.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717045.png" /> be the family of images of the curves in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717046.png" /> under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717047.png" />. If the modulus problem is defined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717048.png" /> and the modulus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717049.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717050.png" />, then the modulus problem is also defined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717052.png" />.
+
Theorem 1. Let $  R $
 +
and $  R _ {1} $
 +
be two conformally-equivalent Riemann surfaces, let $  f $
 +
be a univalent conformal mapping of $  R $
 +
onto $  R _ {1} $,  
 +
let $  \Gamma $
 +
be a family of locally rectifiable curves given on $  R $,  
 +
and let $  \Gamma _ {1} $
 +
be the family of images of the curves in $  \Gamma $
 +
under $  f $.  
 +
If the modulus problem is defined for $  \Gamma $
 +
and the modulus of $  \Gamma $
 +
is $  M ( \Gamma ) $,  
 +
then the modulus problem is also defined for $  \Gamma _ {1} $
 +
and $  M ( \Gamma _ {1} ) = M ( \Gamma ) $.
  
 
The following theorem shows that if there is an extremal metric, then it is essentially unique:
 
The following theorem shows that if there is an extremal metric, then it is essentially unique:
  
Theorem 2. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717053.png" /> be a family of locally rectifiable curves on a Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717054.png" />, and suppose that the modulus problem is defined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717055.png" /> and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717056.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717058.png" /> are extremal metrics for this modulus problem, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717059.png" /> everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717060.png" /> except, possibly, on a subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717061.png" /> of measure zero.
+
Theorem 2. Let $  \Gamma $
 +
be a family of locally rectifiable curves on a Riemann surface $  R $,  
 +
and suppose that the modulus problem is defined for $  \Gamma $
 +
and that $  M ( \Gamma ) \neq \infty $.  
 +
If $  \rho _ {1}  ^ {*} ( z)  | dz | $
 +
and $  \rho _ {2}  ^ {*} ( z)  | dz | $
 +
are extremal metrics for this modulus problem, then $  \rho _ {2}  ^ {*} ( z) = \rho _ {1}  ^ {*} ( z) $
 +
everywhere on $  R $
 +
except, possibly, on a subset of $  R $
 +
of measure zero.
  
 
Examples of moduli of families of curves.
 
Examples of moduli of families of curves.
  
1) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717062.png" /> be a rectangle with sides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717063.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717064.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717065.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717066.png" />) be a family of locally rectifiable curves in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717067.png" /> that join the sides of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717068.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717069.png" />). Then
+
1) Let $  D $
 +
be a rectangle with sides $  a $
 +
and $  b $,  
 +
and let $  \Gamma $(
 +
$  \Gamma _ {1} $)  
 +
be a family of locally rectifiable curves in $  D $
 +
that join the sides of length $  a $(
 +
$  b $).  
 +
Then
  
<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/e/e037/e037170/e03717070.png" /></td> </tr></table>
+
$$
 +
M ( \Gamma )  =
 +
\frac{a}{b}
 +
,\ \
 +
M ( \Gamma _ {1} )  =
 +
\frac{b}{a}
 +
.
 +
$$
  
2) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717071.png" /> be the annulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717072.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717073.png" /> be the class of rectifiable Jordan curves in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717074.png" /> that separate the boundary components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717075.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717076.png" /> be the class of locally rectifiable curves in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717077.png" /> that join the boundary components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717078.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717079.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717080.png" />. In both cases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717081.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717082.png" /> are characteristic conformal invariants of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717083.png" />. Hence, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717084.png" /> is called the modulus of the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717085.png" /> for the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717086.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717087.png" /> is called the modulus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717088.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717089.png" />.
+
2) Let $  D $
 +
be the annulus $  r < | z | < 1 $,  
 +
let $  \Gamma $
 +
be the class of rectifiable Jordan curves in $  D $
 +
that separate the boundary components of $  D $
 +
and let $  \Gamma _ {1} $
 +
be the class of locally rectifiable curves in $  D $
 +
that join the boundary components of $  D $.  
 +
Then $  M ( \Gamma ) = (  \mathop{\rm ln}  1 / r ) / 2 \pi $
 +
and $  M ( \Gamma _ {1} ) = 2 \pi / \mathop{\rm ln} ( 1 / r ) $.  
 +
In both cases $  M ( \Gamma ) $
 +
and $  M ( \Gamma _ {1} ) $
 +
are characteristic conformal invariants of $  D $.  
 +
Hence, $  M ( \Gamma ) $
 +
is called the modulus of the domain $  D $
 +
for the class $  \Gamma $
 +
and $  M ( \Gamma _ {1} ) $
 +
is called the modulus of $  D $
 +
for $  \Gamma _ {1} $.
  
There is a well-known connection between the moduli of families of curves under a [[Quasi-conformal mapping|quasi-conformal mapping]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717090.png" /> be a family of curves in some domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717091.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717092.png" /> be the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717093.png" /> under a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717095.png" />-quasi-conformal mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717096.png" />. Then the moduli <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717097.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717098.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e03717099.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170100.png" />, respectively, satisfy the inequality
+
There is a well-known connection between the moduli of families of curves under a [[Quasi-conformal mapping|quasi-conformal mapping]]. Let $  \Gamma $
 +
be a family of curves in some domain $  D $
 +
and let $  \Gamma _ {1} $
 +
be the image of $  \Gamma $
 +
under a $  K $-
 +
quasi-conformal mapping of $  D $.  
 +
Then the moduli $  M ( \Gamma ) $
 +
and $  M ( \Gamma _ {1} ) $
 +
of $  \Gamma $
 +
and $  \Gamma _ {1} $,  
 +
respectively, satisfy the inequality
  
<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/e/e037/e037170/e037170101.png" /></td> </tr></table>
+
$$
 +
K  ^ {-} 1 M ( \Gamma )  \leq  M ( \Gamma _ {1} )  \leq  K M ( \Gamma ) .
 +
$$
  
The generalization of the concept of the modulus to several families of curves turns out to be important in applications. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170102.png" /> be families of locally rectifiable curves on a Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170103.png" /> (as a rule, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170104.png" /> are, respectively, homotopy classes of curves). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170105.png" /> be non-negative real numbers, not all equal to zero, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170106.png" /> be the class of conformally-invariant metrics <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170107.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170108.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170109.png" /> is integrable for every local parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170110.png" /> and such that
+
The generalization of the concept of the modulus to several families of curves turns out to be important in applications. Let $  \Gamma _ {1} \dots \Gamma _ {n} $
 +
be families of locally rectifiable curves on a Riemann surface $  R $(
 +
as a rule, $  \Gamma _ {1} \dots \Gamma _ {n} $
 +
are, respectively, homotopy classes of curves). Let $  \alpha _ {1} \dots \alpha _ {n} $
 +
be non-negative real numbers, not all equal to zero, and let $  P ( \{ \Gamma _ {j} \} , \{ \alpha _ {j} \} ) $
 +
be the class of conformally-invariant metrics $  \rho ( z)  | dz | $
 +
on $  R $
 +
for which $  \rho  ^ {2} ( z) $
 +
is integrable for every local parameter $  z = x + i y $
 +
and such that
  
<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/e/e037/e037170/e037170111.png" /></td> </tr></table>
+
$$
 +
\int\limits _ {\gamma _ {j} } \rho ( z)  | d z |  \geq  \alpha _ {j} \ \
 +
\textrm{ for }  \gamma _ {j} \in \Gamma _ {j} ,\
 +
j = 1 \dots n .
 +
$$
  
If the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170112.png" /> is non-empty, then the modulus problem <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170113.png" /> is said to be defined for the families of curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170114.png" /> and the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170115.png" />. In this case the quantity
+
If the set $  P ( \{ \Gamma _ {j} \} , \{ \alpha _ {j} \} ) $
 +
is non-empty, then the modulus problem $  {\mathcal P} ( \{ \Gamma _ {j} \} , \{ \alpha _ {j} \} ) $
 +
is said to be defined for the families of curves $  \{ \Gamma _ {j} \} $
 +
and the numbers $  \{ \alpha _ {j} \} $.  
 +
In this case the quantity
  
<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/e/e037/e037170/e037170116.png" /></td> </tr></table>
+
$$
 +
M ( \{ \Gamma _ {j} \} , \{ \alpha _ {j} \} )  = \
 +
\inf _ {\rho \in P ( \{ \Gamma _ {j} \} ,\
 +
\{ \alpha _ {j} \} ) } \
 +
{\int\limits \int\limits } _ { R } \rho  ^ {2} ( z)  d x  d y
 +
$$
  
is called the modulus of this problem. If in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170117.png" /> there is a metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170118.png" /> for which
+
is called the modulus of this problem. If in $  P ( \{ \Gamma _ {j} \} , \{ \alpha _ {j} \} ) $
 +
there is a metric $  \rho  ^ {*} ( z)  | dz | $
 +
for which
  
<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/e/e037/e037170/e037170119.png" /></td> </tr></table>
+
$$
 +
{\int\limits \int\limits } _ { R } [ \rho  ^ {*} ( z) ]  ^ {2} \
 +
d x  d y  = M ( \{ \Gamma _ {j} \} , \{ \alpha _ {j} \} ) ,
 +
$$
  
then this metric is called an extremal metric for the modulus problem <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170120.png" />.
+
then this metric is called an extremal metric for the modulus problem $  {\mathcal P} ( \{ \Gamma _ {j} \} , \{ \alpha _ {j} \} ) $.
  
The modulus problem defined in this way is also a conformal invariant. For such moduli a uniqueness theorem analogous to Theorem 2 holds. The existence of an extremal metric for the modulus problem <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170121.png" /> has been proved under fairly general assumptions. The above definition extends to the case of families of curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170122.png" /> on a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170123.png" /> obtained by removing from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170124.png" /> finitely many points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170125.png" />, where the families <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170126.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170127.png" />, consist of closed Jordan curves homotopic on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170128.png" /> to circles of sufficiently small radii and centres at corresponding selected points. Such an extremal-metric problem in conjunction with the above concept of the modulus of a simply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170129.png" /> relative to a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170130.png" /> (see [[Modulus of an annulus|Modulus of an annulus]]) is connected with the theory of [[Capacity|capacity]] of plane sets.
+
The modulus problem defined in this way is also a conformal invariant. For such moduli a uniqueness theorem analogous to Theorem 2 holds. The existence of an extremal metric for the modulus problem $  {\mathcal P} ( \{ \Gamma _ {j} \} , \{ \alpha _ {j} \} ) $
 +
has been proved under fairly general assumptions. The above definition extends to the case of families of curves $  \Gamma _ {1} \dots \Gamma _ {n} $
 +
on a surface $  R _ {1} $
 +
obtained by removing from $  R $
 +
finitely many points $  a _ {1} \dots a _ {N} $,  
 +
where the families $  \Gamma _ {1} \dots \Gamma _ {k} $,  
 +
$  k \leq  n $,  
 +
consist of closed Jordan curves homotopic on $  R _ {1} $
 +
to circles of sufficiently small radii and centres at corresponding selected points. Such an extremal-metric problem in conjunction with the above concept of the modulus of a simply-connected domain $  D $
 +
relative to a point $  a \in D $(
 +
see [[Modulus of an annulus|Modulus of an annulus]]) is connected with the theory of [[Capacity|capacity]] of plane sets.
  
 
Other generalizations and modifications of the concept of the modulus of a family of curves are also known (see [[#References|[6]]]–[[#References|[10]]]). This concept has been extended to the case of curves and surfaces in space. Uniqueness theorems and a number of properties of such moduli have been established, in particular, an analogue of the inequalities
 
Other generalizations and modifications of the concept of the modulus of a family of curves are also known (see [[#References|[6]]]–[[#References|[10]]]). This concept has been extended to the case of curves and surfaces in space. Uniqueness theorems and a number of properties of such moduli have been established, in particular, an analogue of the inequalities
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037170/e037170131.png" />-quasi-conformal mappings in space has been obtained (see [[#References|[9]]] and [[#References|[10]]]).
+
for $  K $-
 +
quasi-conformal mappings in space has been obtained (see [[#References|[9]]] and [[#References|[10]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.V. Ahlfors,  A. Beurling,  "Conformal invariants and function-theoretic null-sets"  ''Acta Math.'' , '''83'''  (1950)  pp. 101–129</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.A. Jenkins,  "Univalent functions and conformal mapping" , Springer  (1958)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.V. Ahlfors,  "Lectures on quasiconformal mappings" , v. Nostrand  (1966)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.A. Jenkins,  "On the existence of certain general extremal metrics"  ''Ann. of Math.'' , '''66''' :  3  (1957)  pp. 440–453</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  G.V. Kuz'mina,  "Moduli of families of curves and quadratic differentials" , Amer. Math. Soc.  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  J. Hersch,  "Longeurs extrémales et théorie des fonctions"  ''Comment. Math. Helv.'' , '''29''' :  4  (1955)  pp. 301–337</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  P.M. Tamrazov,  "A theorem of line integrals for extremal length"  ''Dokl. Akad. Nauk Ukrain. SSSR'' , '''1'''  (1966)  pp. 51–54  ((in Ukrainian; English summary))</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  B. Fuglede,  "Extremal length and functional completion"  ''Acta Math.'' , '''98'''  (1957)  pp. 171–219</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  B.V. Shabat,  "The modulus method in space"  ''Soviet Math. Dokl.'' , '''1''' :  1  (1960)  pp. 165–168  ''Dokl. Akad. Nauk SSSR'' , '''130''' :  6  (1960)  pp. 1210–1213</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  A.V. Sychev,  "Moduli and quasi-conformal mappings in space" , Novosibirsk  (1983)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.V. Ahlfors,  A. Beurling,  "Conformal invariants and function-theoretic null-sets"  ''Acta Math.'' , '''83'''  (1950)  pp. 101–129</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.A. Jenkins,  "Univalent functions and conformal mapping" , Springer  (1958)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.V. Ahlfors,  "Lectures on quasiconformal mappings" , v. Nostrand  (1966)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.A. Jenkins,  "On the existence of certain general extremal metrics"  ''Ann. of Math.'' , '''66''' :  3  (1957)  pp. 440–453</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  G.V. Kuz'mina,  "Moduli of families of curves and quadratic differentials" , Amer. Math. Soc.  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  J. Hersch,  "Longeurs extrémales et théorie des fonctions"  ''Comment. Math. Helv.'' , '''29''' :  4  (1955)  pp. 301–337</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  P.M. Tamrazov,  "A theorem of line integrals for extremal length"  ''Dokl. Akad. Nauk Ukrain. SSSR'' , '''1'''  (1966)  pp. 51–54  ((in Ukrainian; English summary))</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  B. Fuglede,  "Extremal length and functional completion"  ''Acta Math.'' , '''98'''  (1957)  pp. 171–219</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  B.V. Shabat,  "The modulus method in space"  ''Soviet Math. Dokl.'' , '''1''' :  1  (1960)  pp. 165–168  ''Dokl. Akad. Nauk SSSR'' , '''130''' :  6  (1960)  pp. 1210–1213</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  A.V. Sychev,  "Moduli and quasi-conformal mappings in space" , Novosibirsk  (1983)  (In Russian)</TD></TR></table>

Latest revision as of 19:38, 5 June 2020


of a family of curves

A concept which, along with that of the modulus of a family of curves, is a general form of the definition of conformal invariants and lies at the basis of the method of the extremal metric (cf. Extremal metric, method of the).

Let $ \Gamma $ be a family of locally rectifiable curves on a Riemann surface $ R $. The modulus problem is defined for $ \Gamma $ if there is a non-empty class $ P $ of conformally-invariant metrics (cf. Conformally-invariant metric) $ \rho ( z) | d z | $ given on $ R $ such that $ \rho ( z) $ is square integrable in the $ z $- plane for every local uniformizing parameter $ z ( = x + i y ) $ and if

$$ A _ \rho ( R) = {\int\limits \int\limits } _ { R } \rho ^ {2} ( z) d x d y \ \ \textrm{ and } \ L _ \rho ( \Gamma ) = \inf _ {\gamma \in \Gamma } \int\limits _ \gamma \rho ( z) | d z | $$

are not simultaneously equal to $ 0 $ or $ \infty $. (Each of the above integrals is understood as a Lebesgue integral.) In this case the quantity

$$ M ( \Gamma ) = \inf _ {\rho \in P } \ \frac{A _ \rho ( R) }{[ L _ \rho ( \Gamma ) ] ^ {2} } $$

is called the modulus of the family of curves $ \Gamma $. The reciprocal of $ M ( \Gamma ) $ is called the extremal length of the family of curves $ \Gamma $.

The modulus problem for a family of curves is often defined as follows: Let $ P _ {L} $ be the subclass of $ P $ such that for $ \rho \in P _ {L} $ and $ \gamma \in \Gamma $,

$$ \int\limits _ \gamma \rho ( z) | d z | \geq 1 . $$

If the set $ P _ {L} $ is non-empty, then the quantity

$$ M ( \Gamma ) = \inf _ {\rho \in P _ {L} } A _ \rho ( R) $$

is called the modulus of the family $ \Gamma $. If $ P $ is non-empty but $ P _ {L} $ is empty, then $ M ( \Gamma ) $ is assigned the value $ \infty $. It is the latter definition of the modulus that is adopted below.

Let $ \Gamma $ be a family of locally rectifiable curves on a Riemann surface $ R $ for which the modulus problem is defined, and let $ M ( \Gamma ) \neq \infty $. Then every metric from $ P _ {L} $ is an admissible metric for the modulus problem for $ \Gamma $. If in $ P _ {L} $ there is a metric $ \rho ^ {*} ( z) | dz | $ for which

$$ M ( \Gamma ) = \inf _ {\rho \in P _ {L} } A _ \rho ( R) , $$

then this metric is called an extremal metric in the modulus problem for $ \Gamma $.

The fundamental property of the modulus is its conformal invariance.

Theorem 1. Let $ R $ and $ R _ {1} $ be two conformally-equivalent Riemann surfaces, let $ f $ be a univalent conformal mapping of $ R $ onto $ R _ {1} $, let $ \Gamma $ be a family of locally rectifiable curves given on $ R $, and let $ \Gamma _ {1} $ be the family of images of the curves in $ \Gamma $ under $ f $. If the modulus problem is defined for $ \Gamma $ and the modulus of $ \Gamma $ is $ M ( \Gamma ) $, then the modulus problem is also defined for $ \Gamma _ {1} $ and $ M ( \Gamma _ {1} ) = M ( \Gamma ) $.

The following theorem shows that if there is an extremal metric, then it is essentially unique:

Theorem 2. Let $ \Gamma $ be a family of locally rectifiable curves on a Riemann surface $ R $, and suppose that the modulus problem is defined for $ \Gamma $ and that $ M ( \Gamma ) \neq \infty $. If $ \rho _ {1} ^ {*} ( z) | dz | $ and $ \rho _ {2} ^ {*} ( z) | dz | $ are extremal metrics for this modulus problem, then $ \rho _ {2} ^ {*} ( z) = \rho _ {1} ^ {*} ( z) $ everywhere on $ R $ except, possibly, on a subset of $ R $ of measure zero.

Examples of moduli of families of curves.

1) Let $ D $ be a rectangle with sides $ a $ and $ b $, and let $ \Gamma $( $ \Gamma _ {1} $) be a family of locally rectifiable curves in $ D $ that join the sides of length $ a $( $ b $). Then

$$ M ( \Gamma ) = \frac{a}{b} ,\ \ M ( \Gamma _ {1} ) = \frac{b}{a} . $$

2) Let $ D $ be the annulus $ r < | z | < 1 $, let $ \Gamma $ be the class of rectifiable Jordan curves in $ D $ that separate the boundary components of $ D $ and let $ \Gamma _ {1} $ be the class of locally rectifiable curves in $ D $ that join the boundary components of $ D $. Then $ M ( \Gamma ) = ( \mathop{\rm ln} 1 / r ) / 2 \pi $ and $ M ( \Gamma _ {1} ) = 2 \pi / \mathop{\rm ln} ( 1 / r ) $. In both cases $ M ( \Gamma ) $ and $ M ( \Gamma _ {1} ) $ are characteristic conformal invariants of $ D $. Hence, $ M ( \Gamma ) $ is called the modulus of the domain $ D $ for the class $ \Gamma $ and $ M ( \Gamma _ {1} ) $ is called the modulus of $ D $ for $ \Gamma _ {1} $.

There is a well-known connection between the moduli of families of curves under a quasi-conformal mapping. Let $ \Gamma $ be a family of curves in some domain $ D $ and let $ \Gamma _ {1} $ be the image of $ \Gamma $ under a $ K $- quasi-conformal mapping of $ D $. Then the moduli $ M ( \Gamma ) $ and $ M ( \Gamma _ {1} ) $ of $ \Gamma $ and $ \Gamma _ {1} $, respectively, satisfy the inequality

$$ K ^ {-} 1 M ( \Gamma ) \leq M ( \Gamma _ {1} ) \leq K M ( \Gamma ) . $$

The generalization of the concept of the modulus to several families of curves turns out to be important in applications. Let $ \Gamma _ {1} \dots \Gamma _ {n} $ be families of locally rectifiable curves on a Riemann surface $ R $( as a rule, $ \Gamma _ {1} \dots \Gamma _ {n} $ are, respectively, homotopy classes of curves). Let $ \alpha _ {1} \dots \alpha _ {n} $ be non-negative real numbers, not all equal to zero, and let $ P ( \{ \Gamma _ {j} \} , \{ \alpha _ {j} \} ) $ be the class of conformally-invariant metrics $ \rho ( z) | dz | $ on $ R $ for which $ \rho ^ {2} ( z) $ is integrable for every local parameter $ z = x + i y $ and such that

$$ \int\limits _ {\gamma _ {j} } \rho ( z) | d z | \geq \alpha _ {j} \ \ \textrm{ for } \gamma _ {j} \in \Gamma _ {j} ,\ j = 1 \dots n . $$

If the set $ P ( \{ \Gamma _ {j} \} , \{ \alpha _ {j} \} ) $ is non-empty, then the modulus problem $ {\mathcal P} ( \{ \Gamma _ {j} \} , \{ \alpha _ {j} \} ) $ is said to be defined for the families of curves $ \{ \Gamma _ {j} \} $ and the numbers $ \{ \alpha _ {j} \} $. In this case the quantity

$$ M ( \{ \Gamma _ {j} \} , \{ \alpha _ {j} \} ) = \ \inf _ {\rho \in P ( \{ \Gamma _ {j} \} ,\ \{ \alpha _ {j} \} ) } \ {\int\limits \int\limits } _ { R } \rho ^ {2} ( z) d x d y $$

is called the modulus of this problem. If in $ P ( \{ \Gamma _ {j} \} , \{ \alpha _ {j} \} ) $ there is a metric $ \rho ^ {*} ( z) | dz | $ for which

$$ {\int\limits \int\limits } _ { R } [ \rho ^ {*} ( z) ] ^ {2} \ d x d y = M ( \{ \Gamma _ {j} \} , \{ \alpha _ {j} \} ) , $$

then this metric is called an extremal metric for the modulus problem $ {\mathcal P} ( \{ \Gamma _ {j} \} , \{ \alpha _ {j} \} ) $.

The modulus problem defined in this way is also a conformal invariant. For such moduli a uniqueness theorem analogous to Theorem 2 holds. The existence of an extremal metric for the modulus problem $ {\mathcal P} ( \{ \Gamma _ {j} \} , \{ \alpha _ {j} \} ) $ has been proved under fairly general assumptions. The above definition extends to the case of families of curves $ \Gamma _ {1} \dots \Gamma _ {n} $ on a surface $ R _ {1} $ obtained by removing from $ R $ finitely many points $ a _ {1} \dots a _ {N} $, where the families $ \Gamma _ {1} \dots \Gamma _ {k} $, $ k \leq n $, consist of closed Jordan curves homotopic on $ R _ {1} $ to circles of sufficiently small radii and centres at corresponding selected points. Such an extremal-metric problem in conjunction with the above concept of the modulus of a simply-connected domain $ D $ relative to a point $ a \in D $( see Modulus of an annulus) is connected with the theory of capacity of plane sets.

Other generalizations and modifications of the concept of the modulus of a family of curves are also known (see [6][10]). This concept has been extended to the case of curves and surfaces in space. Uniqueness theorems and a number of properties of such moduli have been established, in particular, an analogue of the inequalities

for $ K $- quasi-conformal mappings in space has been obtained (see [9] and [10]).

References

[1] L.V. Ahlfors, A. Beurling, "Conformal invariants and function-theoretic null-sets" Acta Math. , 83 (1950) pp. 101–129
[2] J.A. Jenkins, "Univalent functions and conformal mapping" , Springer (1958)
[3] L.V. Ahlfors, "Lectures on quasiconformal mappings" , v. Nostrand (1966)
[4] J.A. Jenkins, "On the existence of certain general extremal metrics" Ann. of Math. , 66 : 3 (1957) pp. 440–453
[5] G.V. Kuz'mina, "Moduli of families of curves and quadratic differentials" , Amer. Math. Soc. (1982) (Translated from Russian)
[6] J. Hersch, "Longeurs extrémales et théorie des fonctions" Comment. Math. Helv. , 29 : 4 (1955) pp. 301–337
[7] P.M. Tamrazov, "A theorem of line integrals for extremal length" Dokl. Akad. Nauk Ukrain. SSSR , 1 (1966) pp. 51–54 ((in Ukrainian; English summary))
[8] B. Fuglede, "Extremal length and functional completion" Acta Math. , 98 (1957) pp. 171–219
[9] B.V. Shabat, "The modulus method in space" Soviet Math. Dokl. , 1 : 1 (1960) pp. 165–168 Dokl. Akad. Nauk SSSR , 130 : 6 (1960) pp. 1210–1213
[10] A.V. Sychev, "Moduli and quasi-conformal mappings in space" , Novosibirsk (1983) (In Russian)
How to Cite This Entry:
Extremal length. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Extremal_length&oldid=13112
This article was adapted from an original article by G.V. Kuz'mina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article