Learning higher category theory is a bit like eating a far too big pizza. It's hard to stomach at first, but in the end it's just a pizza, just like "normal" category theory. Also, it's a pineapple-pizza.
/mg/
Caleb Allen
Jonathan Ward
TURN ON CNN
A proof of the Jacobian conjecture
arxiv.org
>arxiv.org
arxiv.org
Lucas Hall
Thanks! Is there some general routine for doing so? Like if you have a 100 degree polynomial with coefficients which don't all eliminate modulo that prime, is there some way of algrebraically proving that, or do you need to check manually?
Hudson James
I feel like I'm missing something really obvious here, but could someone please give me some intuition as to why you'd want to take the projection of a vector connecting two points onto a distance vector or normal vector, when you're trying to do stuff like calculate distance between a point and a line or a point and a plane? I understand how to do it but not really why. My course material focuses a lot on computation and seems to have a pretty treatment of geometric properties of dot and cross products in general.
Ethan James
pretty poor treatment*
Parker Reyes
direction vector even
Jose Wright
See pic.
If you don't understand something, ask.
Julian King
As mentioned, the projection is used in problems of minimizing distances. For example, if you have a vector line and a point outside that line, you find the distance between the point and the line by projecting the point onto the line.
But the projection also has other important properties. For example, projections give rise to what is known as Fourier coefficients which are very important for many reasons, one being that it is possible to prove that with Fourier coefficients one can find the "best" approximation of a vector under certain conditions.
Another important example is that other "geometric" transformations can be written in terms of the projection. For example, the reflection transformation. If you have a vector line and you want to reflect a point relative to that line then what do you do? You take your vector and project it onto the line. Then you move the vector in that same distance and direction so that it ends up on the other "side" of the line. And that is your reflection, in terms of the projection. And what is the reflection? One the first isometries that one can study in a vector space. And then the study of isometries can lead to elegant proofs of various inequalities, for example.
Justin Allen
The great thing is that for a given prime, checking whether a polynomial is irreducible mod that prime is a finite check. I'd usually just have magma do it.
James Sanchez
Probability theory is not statistics.