MathOverflow

Let $U$ be a connected open subset in $\mathbb{R^n}$. Let $f: U \rightarrow U$ be a differentiable projection, i.e. $f\circ f = f$. It's well-known that $f(U)$ is a submanifold of $U$ (Henri Cartan, Sur les rétractions d’une variété.(1986). My question...

Let $X$ be a measurable space and let $P$ be a probability distribution on $X \times \{\pm 1\}$. Let $F$ be a function class on $X$, i.e., a collection of (measurable) functions from $X$ to $\mathbb R$. Fix $\alpha \in \mathbb R$ and $\beta > 0$...

Is there an example of an irreducible polynomial $f(x) \in \mathbb{Q}[x]$ with a real root expressible in terms of real radicals and another real root not expressible in terms of real radicals?

This was originally asked at MSE without success.
Let $$PA_{bd}=\{\varphi: PA\vdash\forall^\infty n([n]\models\varphi)\},$$ where $[n]=\{0,1,...,n\}$ (with $+$ and $\times$ interpreted as $$a+^{[n]}b=\min\{a+b, n\},\quad a\times^{[n]}b=\min\{a\times b, n\}$$ so that this actually makes sense - we could...