Some Aspects of the Numerical Analysis of a Fractional Duffing Oscillator with a Fractional Variable Order Derivative of the Riemann-Liouville Type

In this paper, we consider some aspects of the numerical analysis of the mathematical model of fractional Duffing with a derivative of variable fractional order of the Riemann-Liouville type. Using numerical methods: an explicit finite-difference scheme based on the Grunwald-Letnikov and Adams-Bashford-Moulton approximations (predictor-corrector), the proposed numerical model is found. These methods have been verified with a test case. It is shown that the predictor-corrector method has a faster convergence than the method according to the explicit finite-difference scheme. For these schemes, using Runge's rule, estimates of the computational accuracy were made, which tended to unity with an increase in the number of calculated grid nodes

Signatures of Majorana Kramers pairs in superconductor-Luttinger liquid and superconductor-quantum dot-normal lead junctions

Time-reversal invariant topological superconductors are characterized by the presence of Majorana Kramers pairs localized at defects. One of the transport signatures of Majorana Kramers pairs is the quantized differential conductance of $4e^2/h$ when such a one-dimensional superconductor is coupled to a normal-metal lead. The resonant Andreev reflection, responsible for this phenomenon, can be understood as the boundary condition change for lead electrons at low energies. In this paper, we study the stability of the Andreev reflection fixed point with respect to electron-electron interactions in the Luttinger liquid. We first calculate the phase diagram for the Luttinger liquid-Majorana Kramers pair junction and show that its low-energy properties are determined by Andreev reflection scattering processes in the spin-triplet channel, i.e. the corresponding Andreev boundary conditions are similar to that in a spin-triplet superconductor - normal lead junction. We also study here a quantum dot coupled to a normal lead and a Majorana Kramers pair and investigate the effect of local repulsive interactions leading to an interplay between Kondo and Majorana correlations. Using a combination of renormalization group analysis and slave-boson mean-field theory, we show that the system flows to a new fixed point which is controlled by the Majorana interaction rather than the Kondo coupling. This Majorana fixed point is characterized by correlations between the localized spin and the fermion parity of each spin sector of the topological superconductor. We investigate the stability of the Majorana phase with respect to Gaussian fluctuations.Comment: 26 pages, 8 figure

Fast polarization insensitive optical shutters using dual frequency liquid crystals

Most of the existing displays and optical shutters based on liquid crystals work in combination with linear polarizers. This implies that often more than half of the light is lost due to optical loss in the polarizers and/or the fact that the incoming light is unpolarized. For a number of shutter and filter applications it is important to have a high transmission, while it is not necessary to have a very high contrast. When considering nematic liquid crystals for use in fast optical shutters or filters, a number of possibilities exist. Dual-frequency liquid crystals offer faster switching possibilities because they can be switched from one state to another with a low frequency voltage and switching back can be achieved with the aid of a high frequency voltage. One of the limiting factors for the switching speed of dual-frequency nematics is the appearance of backflow. As in vertically aligned nematic devices, a certain threshold voltage exists above which the switching speed increases drastically [1]. Above the backflow threshold, the liquid crystal ends up in a meta-stable twisted orientation as shown in the figure below