Recent work of Colbrook and Stepaniants produced the first counterexamples to the planar Pompeiu and Schiffer conjectures and introduced the conformal fixed-disc, disk-polynomial, and validated-tail machinery used here. By adapting this framework to the complementary Dirichlet endpoint, we disprove the unrestricted pla...
Matthew J. Colbrook, Siavash Sadeghi, George Stepaniants· 0 citations
The representation number of a graph is the least positive integer $k$ for which its vertices can be arranged in a word, each occurring $k$ times, so that two distinct letters alternate precisely when the corresponding vertices are adjacent. We prove that every bipartite graph on $N\ge9$ vertices has representation num...
Matthew J. Colbrook, Catherine Drysdale· 0 citations
A bounded simply connected noncircular domain of $\Omega$ with real-analytic Jordan boundary and a nonconstant function $u$ such that $u$ is a Neumann eigenfunction which is constant on the boundary, and $\Omega$ is a counterexample to Schiffer's conjecture.
Matthew J. Colbrook, George Stepaniants· 5 citations
Let $\Omega\subset\mathbb{R}^d$, $d\ge2$, be a bounded connected domain with boundary of class $C^{1,\alpha}$, where $0<\alpha<1$. For the adjoint Neumann--Poincar\'e operator $K^*_{\partial\Omega}$, normalised so that its distinguished eigenvalue is $1/2$, let $\lambda_j^+(\Omega)$ denote the upper min--max values on...
Matthew J. Colbrook, Siavash Sadeghi· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
