MathOverflow

Let $B^4 \subset \mathbb R^4$ be the Euclidean unit ball with boundary $S^3$.Consider the Steklov eigenvalue problem$$\begin{cases}\Delta u = 0 & \text{in } \mathbb{B}^4,\\\partial_\nu u = \lambda\, u & \text{on } \mathbb{S}^3,\end{cases}$$where...

Given two finite groups $G,H$ with a homomorphism $\phi:G\to H$ the homomorphism can be injective, surjective or neither. This subdivides the morphisms in the category of finite groups $\mathrm{FinGr}$ effectively into three disjoint parts.
My question: Is there a category theoretical construction which can help us to characterize how $\mathrm{Hom}(G,H)$ splits with respect to this property of its arrows?...

"Let f(x) be a polynomial of degree at least 2 with $f(\mathbb{N})\subset \mathbb{N}$. Then set of natural numbers $n$ such that $f(x)-n$ is reducible has density 0." Is this a true statement? I cannot seem to find a reference anywhere.

Let $k$ be a commutative ring. For a set $M$, denote by $k[M] = \bigoplus_{m \in M} k \cdot m$ the free $k$-module generated by the elements of $M$.In the end, my question will be about constructive mathematics.Free modules can be a little tricky constructively...