A numerical study of variational discretizations of the Camassa–Holm equation

We present two semidiscretizations of the Camassa–Holm equation in periodic domains based on variational formulations and energy conservation. The first is a periodic version of an existing conservative multipeakon method on the real line, for which we propose efficient computation algorithms inspired by works of Camassa and collaborators. The second method, and of primary interest, is the periodic counterpart of a novel discretization of a two-component Camassa–Holm system based on variational principles in Lagrangian variables. Applying explicit ODE solvers to integrate in time, we compare the variational discretizations to existing methods over several numerical examples.

The periodic version of the Da Prato–Grisvard theorem and applications to the bidomain equations with FitzHugh–Nagumo transport

In this article, the periodic version of the classical Da Prato–Grisvard theorem on maximal $${{L}}^p$$ L p -regularity in real interpolation spaces is developed, as well as its extension to semilinear evolution equations. Applying this technique to the bidomain equations subject to ionic transport described by the models of FitzHugh–Nagumo, Aliev–Panfilov, or Rogers–McCulloch, it is proved that this set of equations admits a unique, strongT-periodic solution in a neighborhood of stable equilibrium points provided it is innervated by T-periodic forces.

AN INTEGRATION APPROACH TO THE TOEPLITZ SQUARE PEG PROBLEM

The ‘square peg problem’ or ‘inscribed square problem’ of Toeplitz asks if every simple closed curve in the plane inscribes a (nondegenerate) square, in the sense that all four vertices of that square lie on the curve. By a variety of arguments of a ‘homological’ nature, it is known that the answer to this question is positive if the curve is sufficiently regular. The regularity hypotheses are needed to rule out the possibility of arbitrarily small squares that are inscribed or almost inscribed on the curve; because of this, these arguments do not appear to be robust enough to handle arbitrarily rough curves. In this paper, we augment the homological approach by introducing certain integrals associated to the curve. This approach is able to give positive answers to the square peg problem in some new cases, for instance if the curve is the union of two Lipschitz graphs $f$, $g:[t_{0},t_{1}]\rightarrow \mathbb{R}$ that agree at the endpoints, and whose Lipschitz constants are strictly less than one. We also present some simpler variants of the square problem which seem particularly amenable to this integration approach, including a periodic version of the problem that is not subject to the problem of arbitrarily small squares (and remains open even for regular curves), as well as an almost purely combinatorial conjecture regarding the sign patterns of sums $y_{1}+y_{2}+y_{3}$ for $y_{1},y_{2},y_{3}$ ranging in finite sets of real numbers.

QUANTUM-CHEMISTRY STUDY OF SEMICONDUCTOR SYSTEMS: THE INITIAL OXIDATION OF THE (111)Si-H SURFACE

We propose a well-defined procedure to adapt a Quantum-Chemistry technique for the study of semiconductor structures. The procedure comprises use of a Bloch-periodic version of the standard molecular code and a nanocrystal cluster model. We reparametrize according to this procedure the MNDO/AM1 technique for Si and O and present this new parametrization AM1/Crystal. We apply AM1/Crystal to study oxidation of the hydrogenated silicon(111) surface, at the initial stages, when the absorbed oxygen atoms form isolated surface defects. We obtain self-consistent charge distributions and total energies for optimised geometries of different incorporation sites and configurations, including up to three oxygen atoms. We also obtain vibrational frequencies for the complexes that compare well with available experimental data. Our results confirm that backbond incorporation (Si-O-Si-H) is energetically favored compared to a surface hydroxyl configuration (Si-OH) and indicate that the first subsurface layer incorporation should proceed by island formation in a two-dimensional regime.