Period of superposed cyclic integer rows

[mahch]

Take two rows of respective length m and n:
a₁, a₂, a₃,..., a_m and b₁, b₂, b₃, ..., b_n.

Then produce as follows the generated array G_ai to contain these elements:
a₁, a₁+a₂, a₁+a₂+a₃, ..., a₁+..+a_m,
a₁+..+a_m+a₁, a₁+..+a_m+a₁+a₂, ...

Alike produce the generated array G_bj to contain these elements:
b₁, b₁+b₂, b₁+b₂+b₃, ..., b₁+..+b_n,
b₁+..+b_n+b₁, b₁+..+b_n+b₁+b₂, ...

The numbers a_i and b_j are cumulated cyclically to produce their respective arrays G_ai and G_bj.

Two questions are open to be analyzed (by me) - hope someone has a hint:
1. How to express analitically all numbers contained in G_ai U G_bj as a function.
2. Since the rows a_i and b_j has periods m and n respectively, what is the resulting period of the superposition , ie. the period of G_ai U G_bj?

I appreciate your comment or hint.

 

[ramsey2879]

Take two rows of respective length m and n:
a₁, a₂, a₃,..., a_m and b₁, b₂, b₃, ..., b_n.

Then produce as follows the generated array G_ai to contain these elements:
a₁, a₁+a₂, a₁+a₂+a₃, ..., a₁+..+a_m,
a₁+..+a_m+a₁, a₁+..+a_m+a₁+a₂, ...

Alike produce the generated array G_bj to contain these elements:
b₁, b₁+b₂, b₁+b₂+b₃, ..., b₁+..+b_n,
b₁+..+b_n+b₁, b₁+..+b_n+b₁+b₂, ...

The numbers a_i and b_j are cumulated cyclically to produce their respective arrays G_ai and G_bj.

Two questions are open to be analyzed (by me) - hope someone has a hint:
1. How to express analitically all numbers contained in G_ai U G_bj as a function.
2. Since the rows a_i and b_j has periods m and n respectively, what is the resulting period of the superposition , ie. the period of G_ai U G_bj?

I appreciate your comment or hint.

looks to me like you have an infinite set.

 

[mahch]

Yes sure, the numbers are going infinite, but the indicator function on the set a_i has periodic behaviour. That is what I am aiming at.

To illustrate take the following two simple sets to begin with:
Let a_i = { 7, 4, 7, 4, 7, 12, 3, 12 }
and b_j = { 12, 6, 11, 6, 12, 18, 5, 18 }

whereby the numbers in the set are cumulated round robin, as described in my first post. So the indicator funtion has a 1 on postion 7, 11, 18, 22, 29, 41, 44, 56, 56+7, 56+7+4, etc etc.

The same for the b_js : the indicator has a non-zero (=1) on 12, 18, 29, 35, 47, 65, 70, 88, 88+12, 88+12+6, etc. etc.

So obviously the indicator on both sets a_i and b_j separately, are periodic (periodicity = 8). So the superposition of both periodical sets have a periodicity ( like a discrete Fourier).

The first question would be what that period looks like and how it is calculated. It can be done heuristically with excel sheets, but that is unsatisfactory.
Also, what does the superposed indicatorset looks like.

Having these two superposed, the a third set of c_k would be superposed with the first superposition - and so on.

That is where my (more generalized) initial question arose from.

 

[ramsey2879]

Now that I understand you more, I would think the period will repeat after the indicator element=1 at 8*7*11 since 56 = 8*7 and 88 = 8*11 and have a period as high as 8*(7+11) elements not excluding any duplicate occurrences of the same integer in each set. See if that compares favorably with your spread sheet.

 

[mahch]

Now that I understand you more, I would think the period will repeat after the indicator element=1 at 8*7*11 since 56 = 8*7 and 88 = 8*11 and have a period as high as 8*(7+11) elements not excluding any duplicate occurrences of the same integer in each set. See if that compares favorably with your spread sheet.

Thank you - that covers my results - The Question I am most eager to get answered is: how to 'walk through the ONEs' of the combined indicator function. That is, a function walking me through the ONEs, or better even the other way around, walk me through the ZEROs. I think this should be could be done by a discrete Fourier, but am puzzling how to do this best.

 

[dodo]

To answer part of your question, the 'period' (of the index, in the restricted sense you have used, since the sequence itself is not periodic) of the combination will be http://en.wikipedia.org/wiki/Least_common_multiple" (m,n).

I suppose you have noticed that you only need to describe any particular sequence, say G_a, only up to the length m of the array: if you divide any possible index x by m, to produce a quotient q and a residue r (x = qm+r), then G_ax = q G_am + G_ar.

 

[mahch]

To answer part of your question, the 'period' (of the index, in the restricted sense you have used, since the sequence itself is not periodic) of the combination will be http://en.wikipedia.org/wiki/Least_common_multiple" (m,n).

I suppose you have noticed that you only need to describe any particular sequence, say G_a, only up to the length m of the array: if you divide any possible index x by m, to produce a quotient q and a residue r (x = qm+r), then G_ax = q G_am + G_ar.

absolutely true - this is the periodicity or the modulus. Since multiple 'rows' with their own modulus are superposed, the question still is how to formulate the superposition at number n.