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Lets say you have a set of functions F so that function f1 has a period p1 and so on. How would I go about to find the time t such that all the functions in F are at the start of a new period at t?

Example:

F = {sin(x), sin(2x), sin(0.5x)}
f1 intersects (as multiples of pi): [0, 1, 2, 3, 4]
f2 intersects (as multiples of pi): [0, 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4]
f3 intersects (as multiples of pi): [0, 2, 4]

The only common intersects are 0 and 4 so the period is 4

My initial thought was to take the LCM of the periods, however if the period is a real value I don't really know how to find the LCM.

Any suggestion for how to solve this without produce a set of all the indices that correspond to the start of a period and grabbing the intersection?

Ohunter
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  • The lcm of rational numbers still exists. See https://math.stackexchange.com/questions/44836/rational-numbers-lcm-and-hcf – Chrystomath Aug 18 '20 at 09:12

1 Answers1

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First, note that the periods line up if and only if they are rational multiples of one another. If this condition is satisfied, for example if the periods are $\alpha q_1,\dots,\alpha q_n$ for $\alpha \in \mathbb{R}$ and $q_1,\dots,q_n \in \mathbb{Q}$, then they all line up at time $$ \alpha \cdot\text{lcm}(q_1,\dots,q_n)$$ where the LCM of rational numbers are taken as in the comment above.

Quang Dao
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