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''system in involution, involutive system of partial differential equations''
 
''system in involution, involutive system of partial differential equations''
  
 
A system of first-order partial differential equations
 
A system of first-order partial differential equations
  
<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/i/i052/i052540/i0525401.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
F _ {i} ( x , u , p )  = 0 ,\ \
 +
1 \leq  i \leq  m ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i0525402.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i0525403.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i0525404.png" /> for which all [[Jacobi brackets|Jacobi brackets]] are equal to zero:
+
where $  x = ( x _ {1} \dots x _ {n} ) $,
 +
$  u= u ( x _ {1} \dots x _ {n} ) $,  
 +
$  p = ( p _ {1} \dots p _ {n} ) = ( \partial  u / \partial  x _ {1} \dots \partial  u / \partial  x _ {n} ), $
 +
for which all [[Jacobi brackets|Jacobi brackets]] are equal to zero:
  
<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/i/i052/i052540/i0525405.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
[ F _ {i} , F _ {j} ]  = 0 ,\ \
 +
1 \leq  i , j \leq  m ,
 +
$$
  
identically in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i0525406.png" />. The equations (2) are called the integrability conditions.
+
identically in $  ( x , u , p ) $.  
 +
The equations (2) are called the integrability conditions.
  
This definition is somewhat modified for quasi-linear systems. Suppose that none of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i0525407.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i0525408.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i0525409.png" />, depends on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i05254010.png" />. Then the functions
+
This definition is somewhat modified for quasi-linear systems. Suppose that none of the functions $  \partial  F _ {i} / \partial  p _ {k} $,  
 +
$  1 \leq  i \leq  m $,  
 +
$  1 \leq  k \leq  n $,  
 +
depends on $  p = ( p _ {1} \dots p _ {n} ) $.  
 +
Then the functions
  
<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/i/i052/i052540/i05254011.png" /></td> </tr></table>
+
$$
 +
[ F _ {i} , F _ {j} ] -
 +
F _ {i}
 +
\frac{\partial  F _ {j} }{\partial  u }
 +
+
 +
F _ {j}
 +
\frac{\partial  F _ {i} }{\partial  u }
 +
 
 +
$$
  
 
also have this property. In the class of quasi-linear equations the condition for a system to be in involution (involutional) is defined by the equations
 
also have this property. In the class of quasi-linear equations the condition for a system to be in involution (involutional) is defined by the equations
  
<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/i/i052/i052540/i05254012.png" /></td> </tr></table>
+
$$
 +
[ F _ {i} , F _ {j} ] -
 +
F _ {i}
 +
\frac{\partial  F _ {j} }{\partial  u }
 +
+
 +
F _ {j}
 +
\frac{\partial  F _ {i} }{\partial  u }
 +
  = 0 ,\ \
 +
1 \leq  i , j \leq  m .
 +
$$
  
When the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i05254013.png" /> do not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i05254014.png" />, the definition is the same as the previous one. Sometimes the latter definition is extended to all systems of the form (1).
+
When the $  F _ {i} $
 +
do not depend on $  u $,  
 +
the definition is the same as the previous one. Sometimes the latter definition is extended to all systems of the form (1).
  
 
If the system (1) is linear and homogeneous and is expressed in the form
 
If the system (1) is linear and homogeneous and is expressed in the form
  
<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/i/i052/i052540/i05254015.png" /></td> </tr></table>
+
$$
 +
P _ {i} ( u)  = 0 ,\ \
 +
1 \leq  i \leq  m ,
 +
$$
  
where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i05254016.png" /> are first-order linear differential operators, then being in involution can be defined for it as the commutativity condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i05254017.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i05254018.png" />.
+
where the $  P _ {i} $
 +
are first-order linear differential operators, then being in involution can be defined for it as the commutativity condition $  P _ {i} P _ {j} = P _ {j} P _ {i} $
 +
for all $  1 \leq  i , j \leq  m $.
  
Every system in involution is a [[Complete system|complete system]]. Conversely, if (1) is a complete system and is in normal form, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i05254019.png" /> and
+
Every system in involution is a [[Complete system|complete system]]. Conversely, if (1) is a complete system and is in normal form, that is, $  1 \leq  m \leq  n $
 +
and
  
<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/i/i052/i052540/i05254020.png" /></td> </tr></table>
+
$$
 +
F _ {i} ( x , u , p )  = \
 +
p _ {i} - f _ {i} ( x , u , p _ {m+} 1 \dots p _ {n} ) ,\ \
 +
1 \leq  i \leq  m ,
 +
$$
  
then it is in involution. This enables one to reduce a complete system to a system in involution if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i05254021.png" />, and to solve it by a non-singular transformation with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i05254022.png" /> of the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i05254023.png" />.
+
then it is in involution. This enables one to reduce a complete system to a system in involution if $  m \leq  n $,  
 +
and to solve it by a non-singular transformation with respect to $  m $
 +
of the variables $  p = ( p _ {1} \dots p _ {n} ) $.
  
If the system (1) does not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i05254024.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i05254025.png" />, if the determinant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i05254026.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i05254027.png" /> are solvable from the equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i05254028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i05254029.png" />, then if this system is in involution, the expression
+
If the system (1) does not depend on $  u $,  
 +
if $  m = n $,  
 +
if the determinant $  | \partial  F _ {i} / \partial  p _ {k} | \neq 0 $,  
 +
and if $  p _ {i} = p _ {i} ( x) $
 +
are solvable from the equations $  F _ {i} ( x , p ) = 0 $,
 +
$  1 \leq  i \leq  n $,  
 +
then if this system is in involution, the expression
  
<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/i/i052/i052540/i05254030.png" /></td> </tr></table>
+
$$
 +
\sum _ { i= } 1 ^ { n }
 +
p _ {i} ( x)  d x _ {i}  $$
  
is an exact form. The application of Jacobi's method [[#References|[2]]] for solving systems is involution not dependent on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i05254031.png" /> and consisting of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i05254032.png" /> functional unknown equations, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i05254033.png" />, is based on this. According to this method, the original system is extended to a system in involution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i05254034.png" /> equations with the above properties. The extension proceeds in several stages: each successive system is obtained from the previous one by adding its unknown first integrals. This method also finds application for systems of equations depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i05254035.png" /> (see [[#References|[3]]]).
+
is an exact form. The application of Jacobi's method [[#References|[2]]] for solving systems is involution not dependent on $  u $
 +
and consisting of $  m $
 +
functional unknown equations, $  m < n $,  
 +
is based on this. According to this method, the original system is extended to a system in involution of $  n $
 +
equations with the above properties. The extension proceeds in several stages: each successive system is obtained from the previous one by adding its unknown first integrals. This method also finds application for systems of equations depending on $  u $(
 +
see [[#References|[3]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Carathéodory,  "Calculus of variations and partial differential equations of the first order" , '''1''' , Holden-Day  (1965)  (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C.G.J. Jacobi,  "Nova methodus, aequationes differentiales partiales primi ordinis inter numerum variabilium qeumcunque propositas integrandi"  ''J. Reine Angew. Math.'' , '''60'''  (1862)  pp. 1–181</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E. Coursat,  "Leçons sur l'intégration des équations aux dérivées partielles du premier ordre" , Hermann  (1891)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.M. Gyunter,  "Integrating first-order partial differential equations" , Leningrad-Moscow  (1934)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E. Kamke,  "Differentialgleichungen: Lösungen und Lösungsmethoden" , '''2. Partielle Differentialgleichungen erster Ordnung für die gesuchte Funktion''' , Akad. Verlagsgesell.  (1944)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Carathéodory,  "Calculus of variations and partial differential equations of the first order" , '''1''' , Holden-Day  (1965)  (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C.G.J. Jacobi,  "Nova methodus, aequationes differentiales partiales primi ordinis inter numerum variabilium qeumcunque propositas integrandi"  ''J. Reine Angew. Math.'' , '''60'''  (1862)  pp. 1–181</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E. Coursat,  "Leçons sur l'intégration des équations aux dérivées partielles du premier ordre" , Hermann  (1891)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.M. Gyunter,  "Integrating first-order partial differential equations" , Leningrad-Moscow  (1934)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E. Kamke,  "Differentialgleichungen: Lösungen und Lösungsmethoden" , '''2. Partielle Differentialgleichungen erster Ordnung für die gesuchte Funktion''' , Akad. Verlagsgesell.  (1944)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
For additional references and remarks see also [[Complete system|Complete system]].
 
For additional references and remarks see also [[Complete system|Complete system]].
  
More generally, a system of partial differential equations is defined as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i05254036.png" /> be a fibre manifold, i.e. locally (up to diffeomorphisms) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i05254037.png" /> looks like a standard projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i05254038.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i05254039.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i05254040.png" />-th jet manifold of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i05254041.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i05254042.png" /> be the sheaf of germs of functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i05254043.png" />. Then a system of partial differential equations of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i05254044.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i05254045.png" /> is a locally finitely-generated sheaf of ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i05254046.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i05254047.png" /> restricted to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i05254048.png" />. A solution is a cross section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i05254049.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i05254050.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i05254051.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i05254052.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i05254053.png" />-jet of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i05254054.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i05254055.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052540/i05254056.png" />.
+
More generally, a system of partial differential equations is defined as follows. Let $  \pi : M \rightarrow N $
 +
be a fibre manifold, i.e. locally (up to diffeomorphisms) $  \pi $
 +
looks like a standard projection $  \mathbf R  ^ {n} \times \mathbf R  ^ {m} \rightarrow \mathbf R  ^ {n} $.  
 +
Let $  J  ^ {l} = J  ^ {l} ( \pi ) $
 +
be the $  l $-
 +
th jet manifold of $  \pi $
 +
and let $  {\mathcal O}  ^ {l} $
 +
be the sheaf of germs of functions on $  J  ^ {l} $.  
 +
Then a system of partial differential equations of order $  l $
 +
on $  U \subset  J  ^ {l} $
 +
is a locally finitely-generated sheaf of ideals $  \Phi $
 +
of $  {\mathcal O}  ^ {l} $
 +
restricted to $  U $.  
 +
A solution is a cross section $  f : \pi ( U) \rightarrow U $
 +
such that $  \phi ( j _ {x}  ^ {l} f  ) = 0 $
 +
for all $  \phi \in \Phi $,  
 +
where $  j _ {x}  ^ {l} f $
 +
is the $  l $-
 +
jet of $  f $
 +
at $  x $
 +
in $  \pi ( U) $.
  
 
For a discussion of involutiveness and completeness of systems of partial differential equations in this setting see [[#References|[a1]]].
 
For a discussion of involutiveness and completeness of systems of partial differential equations in this setting see [[#References|[a1]]].

Revision as of 22:13, 5 June 2020


system in involution, involutive system of partial differential equations

A system of first-order partial differential equations

$$ \tag{1 } F _ {i} ( x , u , p ) = 0 ,\ \ 1 \leq i \leq m , $$

where $ x = ( x _ {1} \dots x _ {n} ) $, $ u= u ( x _ {1} \dots x _ {n} ) $, $ p = ( p _ {1} \dots p _ {n} ) = ( \partial u / \partial x _ {1} \dots \partial u / \partial x _ {n} ), $ for which all Jacobi brackets are equal to zero:

$$ \tag{2 } [ F _ {i} , F _ {j} ] = 0 ,\ \ 1 \leq i , j \leq m , $$

identically in $ ( x , u , p ) $. The equations (2) are called the integrability conditions.

This definition is somewhat modified for quasi-linear systems. Suppose that none of the functions $ \partial F _ {i} / \partial p _ {k} $, $ 1 \leq i \leq m $, $ 1 \leq k \leq n $, depends on $ p = ( p _ {1} \dots p _ {n} ) $. Then the functions

$$ [ F _ {i} , F _ {j} ] - F _ {i} \frac{\partial F _ {j} }{\partial u } + F _ {j} \frac{\partial F _ {i} }{\partial u } $$

also have this property. In the class of quasi-linear equations the condition for a system to be in involution (involutional) is defined by the equations

$$ [ F _ {i} , F _ {j} ] - F _ {i} \frac{\partial F _ {j} }{\partial u } + F _ {j} \frac{\partial F _ {i} }{\partial u } = 0 ,\ \ 1 \leq i , j \leq m . $$

When the $ F _ {i} $ do not depend on $ u $, the definition is the same as the previous one. Sometimes the latter definition is extended to all systems of the form (1).

If the system (1) is linear and homogeneous and is expressed in the form

$$ P _ {i} ( u) = 0 ,\ \ 1 \leq i \leq m , $$

where the $ P _ {i} $ are first-order linear differential operators, then being in involution can be defined for it as the commutativity condition $ P _ {i} P _ {j} = P _ {j} P _ {i} $ for all $ 1 \leq i , j \leq m $.

Every system in involution is a complete system. Conversely, if (1) is a complete system and is in normal form, that is, $ 1 \leq m \leq n $ and

$$ F _ {i} ( x , u , p ) = \ p _ {i} - f _ {i} ( x , u , p _ {m+} 1 \dots p _ {n} ) ,\ \ 1 \leq i \leq m , $$

then it is in involution. This enables one to reduce a complete system to a system in involution if $ m \leq n $, and to solve it by a non-singular transformation with respect to $ m $ of the variables $ p = ( p _ {1} \dots p _ {n} ) $.

If the system (1) does not depend on $ u $, if $ m = n $, if the determinant $ | \partial F _ {i} / \partial p _ {k} | \neq 0 $, and if $ p _ {i} = p _ {i} ( x) $ are solvable from the equations $ F _ {i} ( x , p ) = 0 $, $ 1 \leq i \leq n $, then if this system is in involution, the expression

$$ \sum _ { i= } 1 ^ { n } p _ {i} ( x) d x _ {i} $$

is an exact form. The application of Jacobi's method [2] for solving systems is involution not dependent on $ u $ and consisting of $ m $ functional unknown equations, $ m < n $, is based on this. According to this method, the original system is extended to a system in involution of $ n $ equations with the above properties. The extension proceeds in several stages: each successive system is obtained from the previous one by adding its unknown first integrals. This method also finds application for systems of equations depending on $ u $( see [3]).

References

[1] C. Carathéodory, "Calculus of variations and partial differential equations of the first order" , 1 , Holden-Day (1965) (Translated from German)
[2] C.G.J. Jacobi, "Nova methodus, aequationes differentiales partiales primi ordinis inter numerum variabilium qeumcunque propositas integrandi" J. Reine Angew. Math. , 60 (1862) pp. 1–181
[3] E. Coursat, "Leçons sur l'intégration des équations aux dérivées partielles du premier ordre" , Hermann (1891)
[4] N.M. Gyunter, "Integrating first-order partial differential equations" , Leningrad-Moscow (1934) (In Russian)
[5] E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden" , 2. Partielle Differentialgleichungen erster Ordnung für die gesuchte Funktion , Akad. Verlagsgesell. (1944)

Comments

For additional references and remarks see also Complete system.

More generally, a system of partial differential equations is defined as follows. Let $ \pi : M \rightarrow N $ be a fibre manifold, i.e. locally (up to diffeomorphisms) $ \pi $ looks like a standard projection $ \mathbf R ^ {n} \times \mathbf R ^ {m} \rightarrow \mathbf R ^ {n} $. Let $ J ^ {l} = J ^ {l} ( \pi ) $ be the $ l $- th jet manifold of $ \pi $ and let $ {\mathcal O} ^ {l} $ be the sheaf of germs of functions on $ J ^ {l} $. Then a system of partial differential equations of order $ l $ on $ U \subset J ^ {l} $ is a locally finitely-generated sheaf of ideals $ \Phi $ of $ {\mathcal O} ^ {l} $ restricted to $ U $. A solution is a cross section $ f : \pi ( U) \rightarrow U $ such that $ \phi ( j _ {x} ^ {l} f ) = 0 $ for all $ \phi \in \Phi $, where $ j _ {x} ^ {l} f $ is the $ l $- jet of $ f $ at $ x $ in $ \pi ( U) $.

For a discussion of involutiveness and completeness of systems of partial differential equations in this setting see [a1].

References

[a1] M. Kuranishi, "Lectures on involutive systems of partial differential equations" , Publ. Soc. Mat. São Paulo (1967)
[a2] E. Cartan, "Les systèmes différentielles extérieurs et leur applications géométriques" , Hermann (1945)
How to Cite This Entry:
Involutional system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Involutional_system&oldid=47430
This article was adapted from an original article by A.P. Soldatov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article