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Let $\Omega\subset\mathbb{R}^N$ be a bounded and open set (so that $\overline{\Omega}$ is compact). Consider a function $f:\overline{\Omega}\to\mathbb{R}$ with $f\in C(\overline{\Omega})$ and $f\in C^1(\Omega)$.
We know that for any $x_0\in\partial\Omega$ the following limit exists $\lim\limits_{x\to x_0\\ x\in\Omega} \nabla f(x)\in \mathbb{R}^N$. Define the function $L:\overline{\Omega}\to \mathbb{R}^N,\ L(x)=\begin{cases} \nabla f(x),\ x\in\Omega\\ \lim\limits...

Is $$\sum_{n=1}^{+\infty}\frac{\lvert \sin(n) \rvert^n}{n}$$irrational ?
This question is inspired by this MO question Is the series $\sum_n|\sin n|^n/n$ convergent? where Terry Tao proved that this series converges by relating it to the irrationality of $\pi$. I believe it is too optimistic to hope for a closed form of this...

Recently, I found myself in a position where I have to explain sheaf theory at various different levels to people with drastically different specializations—mainly to mathematical physicist who have little to no acquaintance with sheaves or, when familiar...

I asked this question on MSE here
In base $n$ for a positive integer $m$, for each digit $d \in \{0, 1, \dots, n-1\}$, let $s_d$ denote the smallest positive integer such that $d$ appears in the base-$n$ expansion of the product $s_d \cdot m$. The value $a_{n,m}$ is defined as the maximum...