I've had a few neat thoughts to share regarding musical instrument and note detection (regarding Hilbert spaces. No, I'm not joking ;)), but I figure someone's already tried it out. So instead, here's a useless problem where a neat flash of intuition came in handy. They say intuitive flashes are more common when the mind is calm; I'm not sure if this is such an example, but it felt like one.

Consider the wave formed by the sum of sine waves of differing amplitudes and offsets, but the same frequency:

f(x) = a1 * sin (vx + k1) + a2 * sin (vx + k2) ...Does it have multiple peaks (and troughs) per period, or just one? With the usual disclaimer that this may be obvious to everyone but me, here's a quick proof that it's just one:

f''(x) = -v^2 * f(x)Then it occurred to me that it relates to the following situation I was thinking about earlier:

Thus the only inflection points (where f''(x) = 0) are where the wave crosses the x-axis. Hence only one "hump" in-between.

Planets orbiting distant stars are detected by the redshift they induce on the star's light. What can we determine about the number, distance, and masses of planets given just the time series data of the star's redshift?That problem is a bit richer, but the following reduction is useful:

Multiple planets orbiting at the same distance have the same period (not hard to verify: the gravitational acceleration of a body is uniquely determined by its distance from the star, neglecting the pull from other sources), but have different pulls on the star (thus the different amplitudes above). But regardless of their "offsets" around the star, the perturbation of the star will take on what shape? A circle! QED.It feels easier both to relate problems to each other and to use intuition to solve them. I don't know if this will persist, or if it does, whether it will derail my meditation. But it's a nice little break!

(For the sticklers, yes, I left out a few steps: the position being circular implies that the velocity and acceleration vectors are also circular; the acceleration in each dimension is uniquely determined by the position of the planets in those dimensions.)

Note. On second thought, here's a simpler proof: f(x) and f'(x) have the same periodicity. Oops.

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