The following simple-looking inequality for complex numbers in the unit disk generalizes Problem B5 on the Putnam contest 2020: Theorem 1. Let $z_1, z_2, \ldots, z_n$ be $n$ complex numbers such that $\left|z_i\right| \leq 1$ for each $i \in \left\{1,2,\ldots,n\right\}$. Prove that \begin{align}...

$\newcommand\Z{\mathbf{Z}}$Given an integral quadratic form $q$ of signature $(n-1,1)$ and a value $\lambda \in \Z$ is there an algorithm that can determine whether there exist $x\in \Z^n$ such that $q(x) = \lambda$? If yes, are there known implementations? I know that representation of values by...

The strictification theorem for monoidal categories states that every monoidal categorically is monoidally equivalent to a strict monoidal category. Is there a strictification theorem for closed monoidal categories? I expect this to take a form similar to the following. Call a closed monoidal cat...

The $p$-adic field $\mathbb{Q}_p$ has topological basis of open sets of the form $a+p^N \mathbb{Z}_p$ for $0 \leq a \leq p^N-1$ and $N \in \mathbb{Z}$. These are indeed compact open sets. One can define Bernoulli distributions by $$\mu_{B,k}(a+p^N \mathbb{Z}_p)=p^{N(k-1)}B_k \left(\frac{a}{p^N}\r...

Let $ABC$ be arbitrary triangle, $D$, $E$, $F$ are the midpoints of $BC$, $CA$, $AB$ respectively. Define points, segments in the figure below. I am looking for a proof that: $$DE+EF+FD \le (DG+DH+EI+EJ+FJ+FQ).\frac{\varphi}{2}$$ Wher $\varphi=\frac{\sqrt{5}+1}{2}$ the golden ratio. Equality...