Cantor and band spectra for periodic quantum graphs with magnetic fields

We provide an exhaustive spectral analysis of the two-dimensional periodic square graph lattice with a magnetic field. We show that the spectrum consists of the Dirichlet eigenvalues of the edges and of the preimage of the spectrum of a certain discrete operator under the discriminant (Lyapunov function) of a suitable Kronig-Penney Hamiltonian. In particular, between any two Dirichlet eigenvalues the spectrum is a Cantor set for an irrational flux, and is absolutely continuous and has a band structure for a rational flux. The Dirichlet eigenvalues can be isolated or embedded, subject to the choice of parameters. Conditions for both possibilities are given. We show that generically there are infinitely many gaps in the spectrum, and the Bethe-Sommerfeld conjecture fails in this case.Comment: Misprints correcte

Shared genetic risk between eating disorder- and substance-use-related phenotypes:Evidence from genome-wide association studies

First published: 16 February 202

Linking requirements and design data for automated functional evaluation

This paper presents a methodology for automating the evaluation of complex hierarchical designs using black-box testing techniques. Based on an exploration model for design, this methodology generates evaluation tests using a novel semantic graph data model which captures the relationships between the related design and requirements data. Using these relationships, equivalent tests are generated and systematically applied to simulations of the pieces of a modular design and its requirements. These simulations yield two sets of comparable results, enabling evaluation of partial designs of a complex system early in their design process