Surf Wiki
Save to docs
general/real-analysis

From Surf Wiki (app.surf) — the open knowledge base

Interleave sequence

Result of merging two sequences by perfect shuffling

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.
  • 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

References

  1. Strichartz, Robert S.. (2000). "The Way of Analysis". Jones & Bartlett Learning.
  2. Mamoulis, Nikos. (2012). "Spatial Data Management". Morgan & Claypool Publishers.
Info: Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

real-analysis sequences-and-series
Want to explore this topic further?

Ask Mako anything about Interleave sequence — get instant answers, deeper analysis, and related topics.

Research with Mako

Free with your Surf account

Content sourced from Wikipedia, available under CC BY-SA 4.0.

This content may have been generated or modified by AI. CloudSurf Software LLC is not responsible for the accuracy, completeness, or reliability of AI-generated content. Always verify important information from primary sources.

Report