MathOverflow

I just watched the movie Project Hail Mary yesterday, and I have a question about the ending of the movie.
(Spoilers)
At the end, they show the alien nicknamed “Rocky” in a protective clear polyhedral suit that is pretty close-fitting and seemingly quite flexible, allowing him to walk around freely. This is maybe not the most unbelievable aspect of the movie, but it...

For example, for a classical binary classification:
$x \in \mathbb{R}^d$ and $y \in\{0,1\}$
let empirical risk be$R_{\ell}^n(f):=\frac{1}{n} \sum_{i=1}^n \ell\left(f\left(X_i\right), Y_i\right)$

In the course of developing CatDat, we have come across the question: Does the category $\mathbf{Ring}$ of (unital, not necessarily commutative) rings have effective cocongruences?
To unwind the definitions, a cocongruence on $X$ is a pair of morphisms $f, g : X \rightrightarrows E$ which is jointly epimorphic, and such that for each ring $R$, the image of $({-} \circ f, {-} \circ g) : \operatorname{Hom}(E, R) \to \operatorname...

In preparation for a book, a friend and I are trying to cheat our way to the Schoenflies Theorem in $\mathbb{R}^2$ and we have a reduction which seems very assailable. We are hoping to leapfrog right over JCT - in particular, we would like to avoid proving...