A quantitative probabilistic investigation into the accumulation of rounding errors in numerical ODE solution.

We examine numerical rounding errors of some deterministic solvers for systems of ordinary differential equations (ODEs) from a probabilistic viewpoint. We show that the accumulation of rounding errors results in a solution which is inherently random and we obtain the theoretical distribution of the trajectory as a function of time, the step size and the numerical precision of the computer. We consider, in particular, systems which amplify the effect of the rounding errors so that over long time periods the solutions exhibit divergent behaviour. By performing multiple repetitions with different values of the time step size, we observe numerically the random distributions predicted theoretically. We mainly focus on the explicit Euler and fourth order Runge–Kutta methods but also briefly consider more complex algorithms such as the implicit solvers VODE and RADAU5 in order to demonstrate that the observed effects are not specific to a particular method

Vertex corrections in gauge theories for two-dimensional condensed matter systems

We calculate the self-energy of two-dimensional fermions that are coupled to transverse gauge fields, taking two-loop corrections into account. Given a bare gauge field propagator that diverges for small momentum transfers q as 1 / q^{eta}, 1 < eta < 2, the fermionic self-energy without vertex corrections vanishes for small frequencies omega as Sigma (omega) propto omega^{gamma with gamma = {frac{2}{1 + eta}} < 1. We show that inclusion of the leading radiative correction to the fermion - gauge field vertex leads to Sigma (omega) propto omega^{gamma} [ 1 - a_{eta} ln (omega_0 / omega) ], where a_{\eta} is a positive numerical constant and omega_0 is some finite energy scale. The negative logarithmic correction is consistent with the scenario that higher order vertex corrections push the exponent gamma to larger values.Comment: 6 figure

Synchronization of van der Pol oscillators with delayed coupling

The synchronization of self-sustained oscillators such as the van der Pol oscillator is a model for the adjustment of rhythms of oscillating objects due to their weak interaction and has wide applications in natural and technical processes. That these oscillators adjust their frequency or phase to an external forcing or mutually between several oscillators is a phenomenon which can be used in sound synthesis for various purposes. In this paper we focus on the influence of delays on the synchronization properties of these oscillators. As there is no general theory yet on this topic, we mainly present simulation results, together with some background on the non-delayed case. Finally, the theory is also applied in Neukom’s studies 21.1-21.9

The Plasmodium falciparum artemisinin susceptibility-associated AP-2 adaptin mu subunit is clathrin-independent and essential for schizont-maturation v2.0

Proteomics datasets, presented as a set of tables, produced as part of Dr Ryan Henrici's PhD project (completed 2018). Version 2.0 represents an updated version as accepted for publication by mBio in January 2020

Nekhoroshev theorem for the Dirichlet Toda chain

In this work, we prove a Nekhoroshev theorem for the Toda chain with Dirichlet boundary conditions, i.e., fixed ends. The Toda chain is a special case of a Fermi-Pasta-Ulam (FPU) chain, and in view of the unexpected recurrence phenomena observed numerically in these chains, it has been conjectured that theory of perturbed integrable systems could be applied to these chains, especially since the Toda chain has been shown to be a completely integrable system. Whereas various results have already been obtained for the periodic lattice, the Dirichlet chain is more important from the point of view of applications, since the famous numerical experiments have been performed for this type of system. Mathematically, the Dirichlet chain can be treated by exploiting symmetries of the periodic chain. Precisely, by considering the phase space of the Dirichlet chain as an invariant submanifold of the periodic chain, namely the fixed point set of a certain symmetry of the periodic chain, the results obtained for the periodic chain can be used to obtain similar results for the Dirichlet chain. The Nekhoroshev theorem is a perturbation theory result which does not have the probabilistic character of other results such as those of the KAM theorem

Nekhoroshev stability for the Dirichlet Toda lattice

In this work, we prove a Nekhoroshev-type stability theorem for the Toda lattice with Dirichlet boundary conditions, i.e., with fixed ends. The Toda lattice is a member of the family of Fermi-Pasta-Ulam (FPU) chains, and in view of the unexpected recurrence phenomena numerically observed in these chains, it has been a long-standing research aim to apply the theory of perturbed integrable systems to these chains, in particular to the Toda lattice which has been shown to be a completely integrable system. The Dirichlet Toda lattice can be treated mathematically by using symmetries of the periodic Toda lattice. Precisely, by treating the phase space of the former system as an invariant subset of the latter one, namely as the fixed point set of an important symmetry of the periodic lattice, the results already obtained for the periodic lattice can be used to obtain analogous results for the Dirichlet lattice. In this way, we transfer our stability results for the periodic lattice to the Dirichlet lattice. The Nekhoroshev theorem is a perturbation theory result which does not have the probabilistic character of related theorems, and the lattice with fixed ends is more important for applications than the periodic one