Math thread

[samsam]

If your goal is to get an estimate of where the valid configurations are its going to be hard to avoid taking a lot of samples, especially if you dont have any prior knowledge that the problem is linear or convex or something. If you just want to explore the space you can try something like bisection, taking a very coarse grid of points and making it finer as you go. You can be strategic about how you make it finer depending on the density of hits etc.

This feels like the approach that is best for my level of understanding! I gotta be clever so I can run the model several times and make it more refined each time, hmm

[hobartmariner]

It looks like a lot of the usual transmission matrices are projectors onto some basis element (either linear or circular polarization) which means that they should commute since each unique projector is in the null space of each other one. So if the reflection matrices are similar you could have a lot of commutation.

I'd be surprised if at least the matrix A were anything like an ordinary reflection or projection operator, as I+A and 1-A are both invertible. Maybe you're substituting some kind of pseudoinverse for (1+A)^{-1} and (1-A)^{-1}. In ordinary stuff one reflection followed by another corresponds to something like a rotation, although in this Jones setting that might not apply.

[betapersei]

Thats a good point, if they hold the stated equality then the reflection matrices probably have to look different from the transmission matrices.

[slug]

Thats a good point, if they hold the stated equality then the reflection matrices probably have to look different from the transmission matrices.

Still figuring these things out little by little, but apparently due to electronic boundary conditions, the transmission (T) and reflection (R) matrices for a given surfaces are related by T=I+R, where I is the identity. With respect to whether they commute, it seems tricky, because I could imagine setting up a series of lenses that do something different when placed in different orders - ie for transmission, incident light linearly polarized along x through a quarter wave plate produces circularly polarized light, then passing that through a birefringent medium with different indices of refraction along x and y would make it elliptical. But if I flipped the order, I think I end up with circularly polarized light. But there are some experiments I can think of where the order wouldn't matter, so I'm not sure if there's a general commutation relation

edit: realized a quarter wave plate is just a birefringent material with a specific thickness so IDK maybe everything does commute?

[hobartmariner]

Yitang Zhang's comments on Landau-Siegel: https://www.cantorsparadise.com/transla ... 58eeefbb0e

Some funny stuff.