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

z_{i} := {\begin{matrix} x_{i / 2} & if i is even, \\ y_{(i - 1) / 2} & if i is odd. \end{matrix}

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.
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.

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, vol. 21, Morgan & Claypool Publishers, pp. 22–23, ISBN 9781608458325.

This article incorporates material from Interleave sequence on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

[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, vol. 21, Morgan & Claypool Publishers, pp. 22–23, ISBN 9781608458325.