Interleave sequence - Alchetron, The Free Social Encyclopedia

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 , … , be two sequences in S . The interleave sequence is defined to be the sequence x 0 , y 0 , x 1 , y 1 , … . Formally, it is the sequence ( z i ) , i = 0 , 1 , 2 , … given by

z i := { x i / 2 if i is even, y ( i + 1 ) / 2 if i is odd.

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

Interleave sequence Wikipedia

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