MathOverflow

Let $R$ be a ${\rm PID}$ with fraction field $K$, and let $c\in M_n(R)$ be a square matrix. Suppose that $c$ is similar to a Jordan matrix $J_n$ with eigenvalues $0$ over $K$. Can we find an element $a\in R$ such that $c$ is similar to $aJ_n$ over $R...

I guessed the following area formula for a quadrilateral using two opposite sides and four angles: (When $a=AB$ and $c=CD$,)
$$S={a^2\over 2(\cot A+\cot B)}+{c^2\over 2(\cot C+\cot D)}$$
It can be applied under conditions $A+B≠π$, $C+D≠π$, and includes convex, concave, and the particular self-intersecting cases.I'd like to know the scope of applicability and reasons for its validity.Are there any known papers or more elegant ways to...

I was wondering what is known about linear operators on (univariate) polynomials that send polynomials with only positive roots to polynomials with only positive roots. I know Borcea and Brändén arXiv version classified all linear operators on polynomials...

We study the computation of sets of integers using infinitary models of computation.
An Infinite Time Turing Machine (ITTM) is a single-tape model that operates on a space of length $\omega$ with a binary alphabet, using the $\liminf$ rule to determine cell values at limit stages. Its computational power is characterized by $\mathcal...