20 papers
A sharp isoperimetric inequality for the Neumann--Poincaré operator in every dimension
Matthew J. Colbrook, Siavash Sadeghi
Let , , be a bounded connected domain with boundary of class , where . For the adjoint Neumann--Poincaré operator , no…
A computer-assisted counterexample to the planar Berenstein conjecture
Matthew J. Colbrook, Siavash Sadeghi, George Stepaniants
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,…
A Proof of the Forsythe Conjecture for the Two-Step Restarted Conjugate Gradient Method
Matthew J. Colbrook, George Stepaniants, Alex Townsend
Forsythe conjectured in 1968 that the normalized residual directions of restarted conjugate gradients approach a two-cycle. We prove the conjecture for restart length two: unless t…
A computer-assisted counterexample to the planar Pompeiu and Schiffer conjectures
Matthew J. Colbrook, George Stepaniants
The planar Pompeiu problem, originating in 1929, and the associated Schiffer conjecture are long-standing rigidity questions linking rigid-motion integral transforms and Fourier ze…
No Spectral Invisibility for Dissipative Barrier Truncations in Any Dimension
Matthew J. Colbrook, Marco Marletta
The dissipative barrier method suppresses spectral pollution, but whether it can itself conceal genuine spectral points has remained open. Known as the graveyard problem in computa…
Nyström Error Beyond -Matrices: A Minimal Diagonally Dominant Obstruction
Matthew J. Colbrook
We study the nuclear-norm error of a column-selected Nyström approximation to , where is symmetric diagonally dominant and . Our central question is whet…