130 citations · 345 across the 38 of their papers we have counts for
12 papers · 1 filter
Computing Spectral Properties of Differential Operators on Surfaces
Gustav Conradie, Matthew J. Colbrook, Daniel Fortunato
Computing spectra of differential operators on surfaces is crucial in many applications, ranging from medical image analysis to nanoscale quantum structures. However, discretizatio…
Crown graphs maximise the representation number of bipartite graphs
Matthew J. Colbrook, Catherine Drysdale
The representation number of a graph is the least positive integer for which its vertices can be arranged in a word, each occurring times, so that two distinct letters alte…
A counterexample to Kenig's conjecture for the Laplace double-layer operator
Matthew J. Colbrook, Siavash Sadeghi
We disprove Kenig's 1994 spectral-radius conjecture for the double-layer operator in every dimension . For every sufficiently close to , we construct a bounded…
A Complete Resolution of Forsythe's Conjecture for Restarted Conjugate Gradients
Matthew J. Colbrook, George Stepaniants, Alex Townsend
Forsythe's conjecture, published in 1968, asserts that for each restart length , every exact-arithmetic restarted conjugate-gradient iteration on a real symmetric positive defin…
Finiteness and exponential growth of full graph -spectra
Matthew J. Colbrook
We prove that the full spectrum of the graph -Laplacian is finite for every finite graph and every real . This resolves the problem of finiteness of the full graph spectrum…
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 whether…
Ten Digits on a Train: AI-Assisted Verification of Two Eigenvalue Problems
Matthew J. Colbrook
Accurate numerical eigenvalues are often difficult to certify, especially in singular or non-normal settings. This article reports a human--AI collaboration on two such computation…