# Difference between revisions of "Cell"

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− | A term sometimes used to denote a period parallelogram of a [[Double-periodic function|double-periodic function]] | + | <!-- |

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+ | A term sometimes used to denote a period parallelogram of a [[Double-periodic function|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==== | ====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> | ||

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====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 | + | 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|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=15623