Quick Questions: May 25, 2022

Are the appendices of Evans's PDE text sufficient to bring yourself up to speed on the material needed to get through the whole book?
If this is the wrong place for my question I'd be happy to re-post it where appropriate but I'm unsure where is appropriate I hope here.

can some one recommend any literature on modular arithmetic? In particular I've been playing around with sequences of mods for example m1 m2 m3 m4 .... for 3 would be 0103.... as 3 mod 1 = 0 3 mod 2 = 1 ect. it appears the integers have a unique mod sequence there are some nice patterns in the numbers and it's raised some questions for me id like to know more about. in particular I'm not sure how one could define division to get from one modular sequence to another for example

0002106.../002... = 0103.... since that is m1 m2 m3 ect for 6/2 =3 I'm not sure how to arrive from one sequence to the next without converting into integers first or rather the way I do know seems wrong and hackish. I'm also interested if there might be other types of numbers corresponding to other sequences I find the notion intriguing though don't know if it holds any water. I know for instance the negative numbers appear to be uniquely encoded fractions kind of do to "kind of" but I have no notion of what sort of number 01004... might be or 00001000.... ect. perhaps they don't correspond to numbers but I find it interesting I can add subtract and multiply them in a "reasonable" way.  I'm curious if anyone has any literature on this or something similar. are the other sequences corresponding to some type of numbers I've not heard of or is that obviously not the case? my maths education isn't the best but I've enjoyed playing around with the concept and am curious if it goes anywhere known or is just a dead end.
Why does a Compact Self Adjoint operator have only countable eigenvalues? I understand the Spectral Theorem, but conceptually don't get what's special about such operators that they can't have uncountable eigenvalues. Thanks!
I was playing around trying to formalize certain arithmetic properties using just the recursion theorem for a natural numbers object (as axiom) and using only universal properties and commutative diagrams.

I managed to formalize the sum and the product of natural numbers correctly.

But now I wanted to say that every natural number can be either odd or even, and I thought the best way to say this was that N is isomorphic to 2N ⨆ 2N+1, for ⨆ the coproduct.

Now this might not be the best way to say that any natural number can be either odd or even, but my problem is even more basic. I don't know how to say what 2N or 2N+1 are.

I know we can take the functor that makes the following square commute:

N   ->   N
|        |
2^N -> 2^N

where the arrow from above is "multiply by 2" (to take 2N as example); and now we can consider the map 1 -> 2^N that chooses N from 2^N which sent through the morphism from below in the square should be sent to 2N, but it's such a messy definition, I can't really work with this, nor do I know how to write it down and use it in another diagram.

Any help?

Maybe it's just that I'm not seeing how to interpret correctly the "every natural number is either odd or even" proposition.

(Oh and by the way, I was trying to explicitely shy away from the Curry-Howard interpretation.)
Does there exist a sequence of random variables X_n converging to some X in probability but for every subsequence X_n_k of nonzero upper density, P(X_n_k -> X) = 0?
I found this puzzle and wondered if there was any mathematical tool for solving it in a general case:

>In the jungle, there is a rare frog which you know hides in one of four puddles in a line. Every day you can search one puddle, but every night the frog jumps to an adjacent puddle. What is the minimum number of days you need to spend to guarantee you will catch it?

I solved this one using a digraph, but I'm interested in knowing what other kinds of graphs a solution exists for.
Does anyone havea ny good write ups on the graph isomorphism problem? I've been reading about it a little on my own, but a lot of the article I've found tend to be either an overview or dive into the technical details of a particular type of graph isomorphism.

I doubt it will go anywhere, but I was monkeying around with some of the more absurd results you can get with the Axiom of Choice and Infinite Time Turing Machines and I saw something that I believe will let me collapse the polynomial time hierarchy but I need to better understand the graph isomorphism problem. I freely admit I might just be a crank yanking my hog, but either way I'd still like to have a better overall understanding of the Graph Iso problem without having to reference ten dozen papers.
Best textbooks to self-study applications of linear algebra?

I’ve already taken an introductory course in linear algebra and have a background in undergrad math/stats (up to real analysis, PDEs, & numerical analysis), with some programming experience. I’m looking for a textbook that explores the *applications* of linear algebra beyond what a generic introductory course covers. Preferably in topics like statistics, AI/Machine Learning, and computer science. Something that explores connections of linear algebra to derivatives and advanced calculus also sounds interesting, though I suppose that falls under theory.

I’d appreciate any input :)!
by
If a sequence converges in Lp and Lq must the limits be the same?
Textbook recommendations for tensors & manifolds, with some review of topology? I’m ideally looking for a brief treatment of the topic (around 200-300 pages) at the advanced undergrad level. Bonus points if it covers some applications (like topological data analysis).

P.s. I’ve taken classes up to real analysis and linear algebra.

0 like 0 dislike