It's almost impossible to do university level math/physics unless you at least somewhat enjoy the subjects or at the very least have a clear goal you're working towards. If you enjoy a subject and are willing to spend time on it you will eventually become good at it.

If you're a top 10% earner US is and probably always will be one of the best places to live since most policies benefit the top 10%. If you earn above average but below top 10% you are probably in a similar position as people in western europe that are in the same income bracket. Below that and the US is not so great anymore. I don't think there's any reason for this to change in the future either tbh.

The idea is somewhat simple, you simply measure (and in more expensive/sophisticated headphones predict) the noise coming into the ear and use that info to send noise into the ear which exactly counteracts the incoming noise you want to block.

There are several insanely difficult parts to this though:

1. How do we calculate what we need to counteract incoming noise fast enough that we actually counteract it in time? In airpods there may be a couple centimeters between the outer speakers measuring incoming noise and the inner speakers. This gives distance/speed of sound ≈ 3*10^(-2)/343 ≈ 1/10000 seconds, so 100 microseconds. That's not a lot of time to calculate things.

2. The noise being measured at the outer speakers is distorted due to the shape of your ear. This needs to be taken into account or otherwise the inner speakers would not be cancelling out the correct sound.

3. Predicting what noise will be coming and using the prediction instead of actually measuring the noise. This works pretty well for a lot of signal processing purposes (separating noise from music, recreating ECG's when only part of the data is available, or anything else where the input can be considered a signal) but real life noise is not always predictable.

While land isn't exactly cheap today it's probably only going to keep getting more expensive.

When we find solutions to a PDE through separation of variables are we guaranteed to get all possible solutions to the original PDE or could there be missing solutions where the variables are not separable?

For example: PDE P(x,y,z) assumed separable and we find a linear set of solutions in the form X(x)Y(y)Z(z). Could there be a solution in the form A(x,y)Z(z) where the variables x and y are not separable that we can not create through a linear combination of solutions in the form X(x)Y(y)Z(z)?

Probably just because it's possible, it's not like the dna has a goal. It's just pure chance and if any changes increase the chance of dna getting passed on they will persist. It's not like a higher chance of getting cancer in old age has any effect on how many children a person has on average.