Is this a theorem? Every $3$-connected planar graph $G$ may be represented as a tiling of a square by squares, one square per node of $G$, with nodes connected in $G$ corresponding to tangent squares. I ask because the cited AMS Notices paper1 suggests that this might be a theorem, offering ...

If $$ A \subset \Bbb{Z} $$ is such that $xy \in A \implies x \in A, $ or $y \in A$. Then $A$ is either a prime ideal or ? Can we describe all "prime subsets" of $\Bbb{Z}$ that aren't prime ideals in one fell swoop? It's not just a set closed under taking divisors, though those are counted, and s...

Suppose $f : \mathbb{D}\setminus \{0\} \rightarrow \mathbb{C} $ is analytic and $z=0$ is an essential singularity of $f$. Show that the family $\{f_n\} $ defined by $$ f_n(z) = f \left( \frac{z}{2^n} \right), \quad z \in \mathbb{D}\setminus \{0\} $$ is not normal in $\mathbb{D}\setminus \{0\}$. M...

Proposition. Let $f:[0,1]\to\Bbb R$ be a continuously differentiable function such that $f’(x)\ne 0$ for each $x\in [0,1]$. Then $\lim_{n\rightarrow \infty} \int_0^1 |\cos(nf(x))|dx=\tfrac {2}\pi.$ Proof. Since $f’$ is a continuous function on a compact set $[0,1]$, it is uniformly continuous and...

I'm asking for clarity on the method given above. If there is a duplicate post, it is only duplicate if it addresses this. Thx! Let $f(z)$ be a meromorphic function on the complex plane, and suppose there is an integer $m$ such that $f^{-1}(w)$ has at most $m$ points for all $w \in \mathbb{C}$. ...