Interleave sequence

{{Short description|Result of merging two sequences by perfect shuffling}}

In mathematics, an interleave sequence is obtained by merging two sequences via an in shuffle.

Let $S$ be a set, and let $(x_i)$ and $(y_i)$ , $i=0,1,2,\ldots,$ be two sequences in $S.$ The interleave sequence is defined to be the sequence $x_0, y_0, x_1, y_1, \dots$ . Formally, it is the sequence $(z_i), i=0,1,2,\ldots$ given by

: $z_i := \begin{cases} x_{i/2} & \text{ if } i \text{ is even,} \\
y_{(i-1)/2} & \text{ if } i \text{ is odd.} \end{cases}$

Properties

The interleave sequence $(z_i)$ is convergent if and only if the sequences $(x_i)$ and $(y_i)$ are convergent and have the same limit.{{citation|title=The Way of Analysis|first=Robert S.|last=Strichartz|publisher=Jones & Bartlett Learning|year=2000|isbn=9780763714970|page=78|url=https://books.google.com/books?id=Yix09oVvI1IC&pg=PA78}}.
Consider two real numbers a and b greater than zero and smaller than 1. One can interleave the sequences of digits of a and b, which will determine a third number c, also greater than zero and smaller than 1. In this way one obtains an injection from the square {{nowrap|(0, 1) × (0, 1)}} to the interval (0, 1). Different radixes give rise to different injections; the one for the binary numbers is called the Z-order curve or Morton code.{{citation|title=Spatial Data Management|volume=21|series=Synthesis lectures on data management|first=Nikos|last=Mamoulis|publisher=Morgan & Claypool Publishers|year=2012|isbn=9781608458325|pages=22–23|url=https://books.google.com/books?id=6z5grzUcPhoC&pg=PA22}}.

References

Category:Real analysis

Category:Sequences and series