Interleave sequence

In mathematics, an interleave sequence is obtained by merging or shuffling two sequences.

Let $S$ be a set, and let $(x_i)$ and $(y_i)$ , $i=0,1,2,...,$ 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 $(z_i), i=0,1,2,...$ given by

z_i := \left\{\begin{matrix} x_k & \mbox{ if } i=2k \mbox{ is even,}\\ y_k & \mbox{ if } i=2k+1 \mbox{ is odd.} \end{matrix}\right.

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

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

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