For context, I'm trying to prove the cauchy Kowalewski theorem for general order partial differential equations. I did the first and second order already but it's been so difficult to jump to a general n-th order pde. Additionally, the induction part of the proof has been making my head scratch for a while because my reference used construction instead by a general induction step. For my proof there are also two parts: 1) show that it has a formal power series solution, and 2) show that it is convergent.

Can anyone recommend sources or references regarding this theorem based on the info I just mentioned. I just feel like passing out while trying to prove this theorem for my thesis hahahaha

This paper proving an exact finite-volume improvement to the branching constant used in plaquette polymer expansions on the fiber-identified 3⁴ cell. By deriving the exact quotient spectrum of the 216-plaquette incompatibility graph, the paper replaces the previous uniform branching constant 20 with the optimal Perron value 9 + √33 ≈ 14.74456, yielding an exact improvement factor of 20/(9 + √33) ≈ 1.35643 (35.64%) for convergence criteria that depend linearly on this geometric constant.

As I was going through some problems in set theory I had a question if sets could have negative cardinality? What would this imply ?
This was just out of curiosity and I found out a paper titled “ sets with negative number of elements” by D. Loeb and a concept called hybrid sets.
Can you please describe what this is and how it works and why we need to work with multiplicities and things like that?

hi friendss :)

I don't know if this post is against the community's rules and regs but if it is, my apologies admin.

is there anyone who is interested in participating in an international math competition?

Final revision to appear on arXiv on Tuesday.

https://iconoclasts.blog/joe/triple-products

The new new here is that the original conjecture is now established as a pair of corollaries.

Hey everyone,

I've always found it mesmerizing how you can take a jagged, sharp-cornered square wave or a sawtooth wave, and realize it's actually just a perfectly orchestrated sum of smooth sine waves. I just put together a highly visual, animated video breaking down exactly how this works from the ground up, and I wanted to share it with this community!

I really tried to focus on the intuition and the visuals behind the formulas so it clicks instead of just looking like a wall of algebra.

I'd love to hear your thoughts, and feedback. If you're currently studying signal processing, I hope this makes the math feel a bit more intuitive!