Strong convergence rates of probabilistic integrators for ordinary differential equations

Probabilistic integration of a continuous dynamical system is a way of systematically introducing model error, at scales no larger than errors introduced by standard numerical discretisation, in order to enable thorough exploration of possible responses of the system to inputs. It is thus a potentially useful approach in a number of applications such as forward uncertainty quantification, inverse problems, and data assimilation. We extend the convergence analysis of probabilistic integrators for deterministic ordinary differential equations, as proposed by Conrad et al.\ (\textit{Stat.\ Comput.}, 2017), to establish mean-square convergence in the uniform norm on discrete- or continuous-time solutions under relaxed regularity assumptions on the driving vector fields and their induced flows. Specifically, we show that randomised high-order integrators for globally Lipschitz flows and randomised Euler integrators for dissipative vector fields with polynomially-bounded local Lipschitz constants all have the same mean-square convergence rate as their deterministic counterparts, provided that the variance of the integration noise is not of higher order than the corresponding deterministic integrator. These and similar results are proven for probabilistic integrators where the random perturbations may be state-dependent, non-Gaussian, or non-centred random variables.Comment: 25 page

Video in development : filming for rural change

This book is about using video in rural interventions for social change. It gives a glimpse into the many creative ways in which video can be used in rural development activities. Capitalising on experience in this field, the books aims to encourage development professionals to explore the potential of video in development, making it a more coherent, better understood and properly used development tool - in short, filming for rural change

Invariants of differential equations defined by vector fields

We determine the most general group of equivalence transformations for a family of differential equations defined by an arbitrary vector field on a manifold. We also find all invariants and differential invariants for this group up to the second order. A result on the characterization of classes of these equations by the invariant functions is also given.Comment: 13 page

Effects of momentum-dependent nuclear potential on two-nucleon correlation functions and light cluster production in intermediate energy heavy-ion collisions

Using an isospin- and momentum-dependent transport model, we study the effects due to the momentum dependence of isoscalar nuclear potential as well as that of symmetry potential on two-nucleon correlation functions and light cluster production in intermediate energy heavy-ion collisions induced by neutron-rich nuclei. It is found that both observables are affected significantly by the momentum dependence of nuclear potential, leading to a reduction of their sensitivity to the stiffness of nuclear symmetry energy. However, the t/$^{3}$He ratio remains a sensitive probe of the density dependence of nuclear symmetry energy.Comment: 20 pages, 11 figure

Exchange coupling between two ferromagnetic electrodes separated by a graphene nanoribbon

In this study, based on the self-energy method and the total energy calculation, the indirect exchange coupling between two semi-infinite ferromagnetic strips (FM electrodes) separated by metallic graphene nanoribbons (GNRs) is investigated. In order to form a FM/GNR/FM junction, a graphitic region of finite length is coupled to the FM electrodes along graphitic zigzag or armchair interfaces of width $N$. The numerical results show that, the exchange coupling strength which can be obtained from the difference between the total energies of electrons in the ferromagnetic and antiferromagnetic couplings, has an oscillatory behavior, and depends on the Fermi energy and the length of the central region.Comment: 4 pages, 6 figures, International Conference on Theoretical Physics 'Dubna-Nano2008

Deep water periodic waves as Hamiltonian relative equilibria

We use a recently derived KdV-type of equation for waves on deep water to study Stokes waves as relative equilibria. Special attention is given to investigate the cornered Stokes-120 degree wave as a singular solution in the class of smooth steady wave profiles

SiGeC alloy layer formation by high-dose C + implantations into pseudomorphic metastable Ge0.08Si0.92 on Si(100)

Dual-energy carbon implantation (1 × 1016/cm2 at 150 and at 220 keV) was performed on 260-nm-thick undoped metastable pseudomorphic Si(100)/ Ge0.08Si0.92 with a 450-nm-thick SiO2 capping layer, at either room temperature or at 100 °C. After removal of the SiO2 the samples were measured using backscattering/channeling spectrometry and double-crystal x-ray diffractometry. A 150-nm-thick amorphous layer was observed in the room temperature implanted samples. This layer was found to have regrown epitaxially after sequential annealing at 550 °C for 2 h plus at 700 °C for 30 min. Following this anneal, tensile strain, believed to result from a large fraction of substitutional carbon in the regrown layer, was observed. Compressive strain, that presumably arises from the damaged but nonamorphized portion of the GeSi layer, was also observed. This strain was not significantly affected by the annealing treatment. For the samples implanted at 100 °C, in which case no amorphous layer was produced, only compressive strain was observed. For samples implanted at both room temperature and 100 °C, the channelled backscattering yield from the Si substrate was the same as that of the virgin sample

Elliptic CR-manifolds and shear invariant ODE with additional symmetries

We classify the ODEs that correspond to elliptic CR-manifolds with maximal isotropy. It follows that the dimension of the isotropy group of an elliptic CR-manifold can be only 10 (for the quadric), 4 (for the listed examples) or less. This is in contrast with the situation of hyperbolic CR-manifolds, where the dimension can be 10 (for the quadric), 6 or 5 (for semi-quadrics) or less than 4. We also prove that, for all elliptic CR-manifolds with non-linearizable istropy group, except for two special manifolds, the points with non-linearizable isotropy form exactly some complex curve on the manifold