Difference between revisions of "Cell"

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)

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.