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''polycylinder''
 
''polycylinder''
  
 
A region
 
A region
  
<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/p/p073/p073570/p0735701.png" /></td> </tr></table>
+
$$
 +
\Delta  = \Delta ( a = ( a _ {1} \dots a _ {n} ),\
 +
r = ( r _ {1} \dots r _ {n} )) =
 +
$$
  
<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/p/p073/p073570/p0735702.png" /></td> </tr></table>
+
$$
 +
= \
 +
\{ z = ( z _ {1} \dots z _ {n} ) \in \mathbf C  ^ {n} : | z _  \nu  - a _  \nu  | < r _  \nu  , \nu = 1 \dots n \}
 +
$$
  
in a complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073570/p0735703.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073570/p0735704.png" />, which is the topological product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073570/p0735705.png" /> discs
+
in a complex space $  \mathbf C  ^ {n} $,  
 +
$  n \geq  1 $,  
 +
which is the topological product of $  n $
 +
discs
  
<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/p/p073/p073570/p0735706.png" /></td> </tr></table>
+
$$
 +
\Delta  = \Delta _ {1} \times \dots \times \Delta _ {n} ,
 +
$$
  
<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/p/p073/p073570/p0735707.png" /></td> </tr></table>
+
$$
 +
\Delta _  \nu  = \{ z _  \nu  \in \mathbf C : | z _  \nu  - a _  \nu  | < r _  \nu  \} ,\  \nu = 1 \dots n .
 +
$$
  
The point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073570/p0735708.png" /> is the centre of the polydisc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073570/p0735709.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073570/p07357010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073570/p07357011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073570/p07357012.png" />, is its polyradius. With <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073570/p07357013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073570/p07357014.png" /> one obtains the unit polydisc. The distinguished boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073570/p07357015.png" /> is the set
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The point $  a = ( a _ {1} \dots a _ {n} ) \in \mathbf C  ^ {n} $
 +
is the centre of the polydisc $  \Delta $,
 +
$  r = ( r _ {1} \dots r _ {n} ) $,  
 +
$  r _  \nu  > 0 $,  
 +
$  \nu = 1 \dots n $,  
 +
is its polyradius. With $  a = 0 $,  
 +
$  r = ( 1 \dots 1 ) $
 +
one obtains the unit polydisc. The distinguished boundary of $  \Delta $
 +
is the set
  
<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/p/p073/p073570/p07357016.png" /></td> </tr></table>
+
$$
 +
= T( a, r)  = \{ {z \in \mathbf C  ^ {n} } : {
 +
| z _  \nu  - a _  \nu  | = r _  \nu  , \nu = 1 \dots n } \}
 +
,
 +
$$
  
which is a part of its complete topological boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073570/p07357017.png" />. A polydisc is a complete [[Reinhardt domain|Reinhardt domain]].
+
which is a part of its complete topological boundary $  \partial  \Delta $.  
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A polydisc is a complete [[Reinhardt domain|Reinhardt domain]].
  
A natural generalization of the concept of a polydisc is that of a polyregion (polycircular region, generalized polycylinder) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073570/p07357018.png" />, which is the topological product of, in general multiply-connected, regions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073570/p07357019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073570/p07357020.png" />. The boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073570/p07357021.png" /> of a polyregion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073570/p07357022.png" /> consists of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073570/p07357023.png" /> sets of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073570/p07357024.png" />:
+
A natural generalization of the concept of a polydisc is that of a polyregion (polycircular region, generalized polycylinder) $  D = D _ {1} \times \dots \times D _ {n} $,  
 +
which is the topological product of, in general multiply-connected, regions $  D _  \nu  \subset  \mathbf C $,  
 +
$  \nu = 1 \dots n $.  
 +
The boundary $  \Gamma = \partial  D $
 +
of a polyregion $  D $
 +
consists of $  n $
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sets of dimension $  2n - 1 $:
  
<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/p/p073/p073570/p07357025.png" /></td> </tr></table>
+
$$
 
+
\Gamma _  \nu  = \
the common part of which is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073570/p07357026.png" />-dimensional distinguished boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073570/p07357027.png" />:
+
\{ {z \in \mathbf C  ^ {n} } : {z _  \nu  \in \partial  D _  \nu  ,\
 
+
z _  \mu  \in \overline{D}\; _  \mu  , \mu \neq \nu } \}
<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/p/p073/p073570/p07357028.png" /></td> </tr></table>
+
,\ \
 +
\nu = 1 \dots n,
 +
$$
  
 +
the common part of which is the  $  n $-
 +
dimensional distinguished boundary of  $  D $:
  
 +
$$
 +
T  =  \partial  D _ {1} \times \dots \times \partial  D _ {n}  = \
 +
\{ {z \in \mathbf C  ^ {n} } : {
 +
z _  \nu  \in \partial  D _  \nu  , \nu = 1 \dots n } \}
 +
.
 +
$$
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Rudin,  "Function theory in polydiscs" , Benjamin  (1969)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Rudin,  "Function theory in polydiscs" , Benjamin  (1969)</TD></TR></table>

Latest revision as of 08:06, 6 June 2020


polycylinder

A region

$$ \Delta = \Delta ( a = ( a _ {1} \dots a _ {n} ),\ r = ( r _ {1} \dots r _ {n} )) = $$

$$ = \ \{ z = ( z _ {1} \dots z _ {n} ) \in \mathbf C ^ {n} : | z _ \nu - a _ \nu | < r _ \nu , \nu = 1 \dots n \} $$

in a complex space $ \mathbf C ^ {n} $, $ n \geq 1 $, which is the topological product of $ n $ discs

$$ \Delta = \Delta _ {1} \times \dots \times \Delta _ {n} , $$

$$ \Delta _ \nu = \{ z _ \nu \in \mathbf C : | z _ \nu - a _ \nu | < r _ \nu \} ,\ \nu = 1 \dots n . $$

The point $ a = ( a _ {1} \dots a _ {n} ) \in \mathbf C ^ {n} $ is the centre of the polydisc $ \Delta $, $ r = ( r _ {1} \dots r _ {n} ) $, $ r _ \nu > 0 $, $ \nu = 1 \dots n $, is its polyradius. With $ a = 0 $, $ r = ( 1 \dots 1 ) $ one obtains the unit polydisc. The distinguished boundary of $ \Delta $ is the set

$$ T = T( a, r) = \{ {z \in \mathbf C ^ {n} } : { | z _ \nu - a _ \nu | = r _ \nu , \nu = 1 \dots n } \} , $$

which is a part of its complete topological boundary $ \partial \Delta $. A polydisc is a complete Reinhardt domain.

A natural generalization of the concept of a polydisc is that of a polyregion (polycircular region, generalized polycylinder) $ D = D _ {1} \times \dots \times D _ {n} $, which is the topological product of, in general multiply-connected, regions $ D _ \nu \subset \mathbf C $, $ \nu = 1 \dots n $. The boundary $ \Gamma = \partial D $ of a polyregion $ D $ consists of $ n $ sets of dimension $ 2n - 1 $:

$$ \Gamma _ \nu = \ \{ {z \in \mathbf C ^ {n} } : {z _ \nu \in \partial D _ \nu ,\ z _ \mu \in \overline{D}\; _ \mu , \mu \neq \nu } \} ,\ \ \nu = 1 \dots n, $$

the common part of which is the $ n $- dimensional distinguished boundary of $ D $:

$$ T = \partial D _ {1} \times \dots \times \partial D _ {n} = \ \{ {z \in \mathbf C ^ {n} } : { z _ \nu \in \partial D _ \nu , \nu = 1 \dots n } \} . $$

Comments

References

[a1] W. Rudin, "Function theory in polydiscs" , Benjamin (1969)
How to Cite This Entry:
Polydisc. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Polydisc&oldid=19171
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article