MathOverflow

Let $Z={U}^\top {V}$ where ${U}$ and ${V}$ are uniformly distributed $\mathbb{S}^{p-1}$. It is known that $Z$ has even moments given by\begin{align*}\mathbb{E} Z^{2m} = \frac{(2m-1)!!}{p(p+2)\cdots(p+2m-2)}\end{align*}I am wondering if for a universal...

This question arose from quantum physics research (e.g., Quantum Rep. 2022, 4(4), 486-508)
I have an overdetermined system of partial differential equations for five real unknown functions of four independent variables, and I want to eliminate one of them. I would greatly appreciate if you could suggest a specific approach for my problem or...

Given a prime $p$ and by Dirichlet a prime $q = k\cdot p+1$ - minimal of this form -,does then the number $k = (q-1)/p$ have only prime divisors $< p$?What does the research literature say for this question?

Let $f: \mathbb{R}^+\to \mathbb{R}^+, \ x\mapsto e^x-1$ and $g: \mathbb{R}^+\to \mathbb{R}^+, \ x\mapsto x^2$.
Is the group generated by $f$ and $g$ under composition free? (That is, no non-trivial reduced word generated by $f, g, f^{-1}, g^{-1}$ fall to the identity function $\mathrm{id}:x\mapsto x$) By instinct it's true, but I can't figure out the right path...