We investigate the growth of the polynomial and multilinear Hardy--Littlewood inequalities. Analytical and numerical approaches are performed and, in particular, among other results, we show that a simple application of the best known constants of th
e Clarkson inequality improves a recent result of Araujo et al. We also obtain the optimal constants of the generalized Hardy--Littlewood inequality in some special cases.
A nonlinear PDE featuring flux limitation effects together with those of the porous media equation (nonlinear Fokker-Planck) is presented in this paper. We analyze the balance of such diverse effects through the study of the existence and qualitative
behavior of some admissible patterns, namely traveling wave solutions, to this singular reaction-difusion equation. We show the existence and qualitative behavior of different types of traveling waves: classical profiles for wave speeds high enough, and discontinuous waves that are reminiscent of hyperbolic shock waves when the wave speed lowers below a certain threshold. Some of these solutions are of particular relevance as they provide models by which the whole solution (and not just the bulk of it, as it is the case with classical traveling waves) spreads through the medium with finite speed.
Twisted bi-layer graphene (tBLG) has recently attracted interest due to the peculiar electrical properties that arise from its random rotational configurations. Our experiments on CVD-grown graphene from Cu foil and transferred onto Si substrates, wi
th an oxide layer of 100 nm, reveal naturally-produced bi-layer graphene patches which present different colorations when shined with white light. In particular yellow-, pink- and blue- colored areas are evidenced. Combining optical microscopy, Raman spectroscopy and transmission electron microscopy we have been able to assign these colorations to ranges of rotational angles between the two graphene layers. Optical contrast simulations have been carried out, proving that the observation of the different colorations is due to the angle-dependent electronic properties of tBLG combined with the reflection that results from the layered structure tBLG / 100 nm-thick SiO2 / Si. Our results could lead the way to an easy selective identification of bi-layer graphene merely through the observation on an optical microscope.
Bi-layer graphene with a twist angle theta between the layers generates a superlattice structure known as Moir{e} pattern. This superlattice provides a theta-dependent q wavevector that activates phonons in the interior of the Brillouin zone. Here we
show that this superlattice-induced Raman scattering can be used to probe the phonon dispersion in twisted bi-layer graphene (tBLG). The effect reported here is different from the broadly studied double-resonance in graphene-related materials in many aspects, and despite the absence of stacking order in tBLG, layer breathing vibrations (namely ZO phonons) are observed.
A novel approach to study transmission through waveguides in terms of optical streamlines is presented. This theoretical framework combines the computational performance of beam propagation methods with the possibility to monitor the passage of light
through the guiding medium by means of these sampler paths. In this way, not only the optical flow along the waveguide can be followed in detail, but also a fair estimate of the transmitted light (intensity) can be accounted for by counting streamline arrivals with starting points statistically distributed according to the input pulse. Furthermore, this approach allows to elucidate the mechanism leading to energy losses, namely a vortical dynamics, which can be advantageously exploited in optimal waveguide design.
