6 problems

Let $R$ be a Euclidean domain, and suppose that $R$ admits an ultra-Euclidean function. Ultra-Euclidean classification conjecture. Then either $R$ is a field or there is a field…

Let $R$ be a Euclidean domain, let $f:R\setminus\{0\}\to\mathbb{Z}_0^+$ be an ultra-Euclidean function, and let $\widetilde{f}$ denote the associated transformed function. Preserva…

Let $n$ be the ambient dimension, let $u,v\in\mathbb{R}^n$ be distinct points, and let $[u,v]$ denote the line segment joining them. For a parameter $c$, write…

Let $\Omega$ be a bounded domain with piecewise smooth boundary in the $n$-dimensional Euclidean space $\mathbb{R}^{n}$. Let $\Lambda_j$ be the $j$-th Dirichlet eigenvalue of the L…

Let $\Omega$ be a bounded domain in the $n$-dimensional Euclidean space $\mathbb{R}^{n}$. Let $\Lambda_i$ denote the $i$-th Dirichlet eigenvalue of the Laplace operator, and let…

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain with piecewise smooth boundary $\partial\Omega$, and let $\lambda_i$ be the $i$-th Dirichlet eigenvalue. Let…