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### $C^j$-topology considered by Greene and Krantz

My question is about the $C^j$ topology used by Greene and Krantz in their paper "Deformations of Complex Structures, Estimates for the $\bar{\partial}$-equation, and stability of the Bergman ...
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### Semisimple subgroup of Euclidean group

Let $G$ be a closed and connected semisimple subgroup of the Euclidean group $E(n)$ (the group of isometries of $\mathbb R^n$). Can we prove that $G$ is conjugate to a subgroup of $O(n)$?
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### Intuitively, what does a graph Laplacian represent?

Recently I saw an MO post Algebraic graph invariant $\mu(G)$ which links Four-Color-Theorem with Schrödinger operators: further topological characterizations of graphs? that got me interested. ...
Let $f(x),g(x)\in\mathbf{Z}_{q^\alpha}[x]$, $q$ is a prime. If the polynomials $f(x)$ and $g(x)$ have the same factors, is there any relationship between $f(x),g(x)$ ? If $f(x),g(x)\in\mathbf{Z}_n[x]$,...
The Sobolev embedding theorem implies that every function of class $W^{k,p}$ on a reasonable $n$-dimensional domain is continuous if $kp > n$. Cases with $kp=n$ are known as "borderline" ...