Interleave sequence

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 $(y_{i})$, ${\displaystyle i=0,1,2,\ldots ,}$ 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 ${\displaystyle (z_{i}),i=0,1,2,\ldots }$ given by

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

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.^[1]
• 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 (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.^[2]

References

1. Strichartz, Robert S. (2000), The Way of Analysis, Jones & Bartlett Learning, p. 78, ISBN 9780763714970.
2. Mamoulis, Nikos (2012), Spatial Data Management, Synthesis lectures on data management, 21, Morgan & Claypool Publishers, pp. 22–23, ISBN 9781608458325.

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