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A term sometimes used to denote a period parallelogram of a [[Double-periodic function|double-periodic function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021120/c0211201.png" /> whose sides do not contain poles, and which is obtained from a fundamental period parallelogram by translation over a vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021120/c0211202.png" />.
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A term sometimes used to denote a period parallelogram of a [[Double-periodic function|double-periodic function]] $  f $
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whose sides do not contain poles, and which is obtained from a fundamental period parallelogram by translation over a vector $  z _ {0} \in \mathbf C $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.T. Whittaker,  G.N. Watson,  "A course of modern analysis" , Cambridge Univ. Press  (1927)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.T. Whittaker,  G.N. Watson,  "A course of modern analysis" , Cambridge Univ. Press  (1927)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
In addition there are several technical meanings of the word cell in geometry and topology. Thus, in affine geometry the convex hull of a finite set of points is sometimes called a convex cell. A subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021120/c0211203.png" /> of a topological Hausdorff space such that there is a relative homeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021120/c0211204.png" /> is a (topological) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021120/c0211206.png" />-dimensional cell. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021120/c0211207.png" /> is the unit ball, its boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021120/c0211208.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021120/c0211209.png" />-dimensional sphere, and a relative homeomorphism is, of course, a continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021120/c02112010.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021120/c02112011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021120/c02112012.png" /> induces a homeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021120/c02112013.png" />; cf. also [[Cell complex|Cell complex]] and [[Cellular space|Cellular space]]. The phrase unit cell occasionally occurs as a synonym for the ball (disc) of radius one centred at the origin in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021120/c02112014.png" />-dimensional Euclidean space, and any space homeomorphic to it is also sometimes called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021120/c02112016.png" />-dimensional topological cell. Finally, cell is sometimes used to denote the possible locations of entries in a matrix or similar structure, such as a [[Magic square|magic square]] or [[Young diagram|Young diagram]], or as a synonym for a block in a block matrix.
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In addition there are several technical meanings of the word cell in geometry and topology. Thus, in affine geometry the convex hull of a finite set of points is sometimes called a convex cell. A subset $  E $
 +
of a topological Hausdorff space such that there is a relative homeomorphism $  \alpha : ( B  ^ {n} , S ^ {n - 1 } ) \rightarrow ( \overline{E}\; , \partial  E) $
 +
is a (topological) $  n $-
 +
dimensional cell. Here $  B  ^ {n} $
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is the unit ball, its boundary $  \partial  B  ^ {n} = S ^ {n - 1 } $
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is the $  ( n - 1) $-
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dimensional sphere, and a relative homeomorphism is, of course, a continuous mapping $  \alpha : B  ^ {n} \rightarrow \overline{E}\; $
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such that $  \alpha ( S ^ {n - 1 } ) \subset  \partial  E $
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and $  \alpha $
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induces a homeomorphism $  B  ^ {n} \setminus  S ^ {n - 1 } \rightarrow \overline{E}\; \setminus  \partial  E $;  
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cf. also [[Cell complex|Cell complex]] and [[Cellular space|Cellular space]]. The phrase unit cell occasionally occurs as a synonym for the ball (disc) of radius one centred at the origin in $  n $-
 +
dimensional Euclidean space, and any space homeomorphic to it is also sometimes called an $  n $-
 +
dimensional topological cell. Finally, cell is sometimes used to denote the possible locations of entries in a matrix or similar structure, such as a [[Magic square|magic square]] or [[Young diagram|Young diagram]], or as a synonym for a block in a block matrix.

Latest revision as of 16:43, 4 June 2020


A term sometimes used to denote a period parallelogram of a double-periodic function $ f $ whose sides do not contain poles, and which is obtained from a fundamental period parallelogram by translation over a vector $ z _ {0} \in \mathbf C $.

References

[1] E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1927)

Comments

In addition there are several technical meanings of the word cell in geometry and topology. Thus, in affine geometry the convex hull of a finite set of points is sometimes called a convex cell. A subset $ E $ of a topological Hausdorff space such that there is a relative homeomorphism $ \alpha : ( B ^ {n} , S ^ {n - 1 } ) \rightarrow ( \overline{E}\; , \partial E) $ is a (topological) $ n $- dimensional cell. Here $ B ^ {n} $ is the unit ball, its boundary $ \partial B ^ {n} = S ^ {n - 1 } $ is the $ ( n - 1) $- dimensional sphere, and a relative homeomorphism is, of course, a continuous mapping $ \alpha : B ^ {n} \rightarrow \overline{E}\; $ such that $ \alpha ( S ^ {n - 1 } ) \subset \partial E $ and $ \alpha $ induces a homeomorphism $ B ^ {n} \setminus S ^ {n - 1 } \rightarrow \overline{E}\; \setminus \partial E $; cf. also Cell complex and Cellular space. The phrase unit cell occasionally occurs as a synonym for the ball (disc) of radius one centred at the origin in $ n $- dimensional Euclidean space, and any space homeomorphic to it is also sometimes called an $ n $- dimensional topological cell. Finally, cell is sometimes used to denote the possible locations of entries in a matrix or similar structure, such as a magic square or Young diagram, or as a synonym for a block in a block matrix.

How to Cite This Entry:
Cell. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cell&oldid=46291
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article