\epsilon-regularity for systems involving non-local, antisymmetric operators

We prove an epsilon-regularity theorem for critical and super-critical systems with a non-local antisymmetric operator on the right-hand side. These systems contain as special cases, Euler-Lagrange equations of conformally invariant variational functionals as Rivi\`ere treated them, and also Euler-Lagrange equations of fractional harmonic maps introduced by Da Lio-Rivi\`ere. In particular, the arguments presented here give new and uniform proofs of the regularity results by Rivi\`ere, Rivi\`ere-Struwe, Da-Lio-Rivi\`ere, and also the integrability results by Sharp-Topping and Sharp, not discriminating between the classical local, and the non-local situations

LIDAR and beam steering tailored by neuromorphic metasurfaces dipped in a tunable surrounding medium

The control of amplitude, losses and deflection of light with elements of an optical array is of paramount importance for realizing dynamic beam steering for light detection and ranging applications (LIDAR). In this paper, we propose an optical beam steering device, operating at a wavelength of 1550 nm, based on high index material as molybdenum disulfide (MoS2) where the direction of the light is actively controlled by means of liquid crystal. The metasurface have been designed by a deep machine learning algorithm jointed with an optimizer in order to obtain univocal optical responses. The achieved numerical results represent a promising way for the realization of novel LIDAR for future applications with increase control and precision

Hong-Ou-Mandel interference between independent III-V on silicon waveguide integrated lasers

The versatility of silicon photonic integrated circuits has led to a widespread usage of this platform for quantum information based applications, including Quantum Key Distribution (QKD). However, the integration of simple high repetition rate photon sources is yet to be achieved. The use of weak-coherent pulses (WCPs) could represent a viable solution. For example, Measurement Device Independent QKD (MDI-QKD) envisions the use of WCPs to distill a secret key immune to detector side channel attacks at large distances. Thus, the integration of III-V lasers on silicon waveguides is an interesting prospect for quantum photonics. Here, we report the experimental observation of Hong-Ou-Mandel interference with 46\pm 2% visibility between WCPs generated by two independent III-V on silicon waveguide integrated lasers. This quantum interference effect is at the heart of many applications, including MDI-QKD. Our work represents a substantial first step towards an implementation of MDI-QKD fully integrated in silicon, and could be beneficial for other applications such as standard QKD and novel quantum communication protocols.Comment: 5 pages, 3 figure

An Optimal Execution Problem with Market Impact

We study an optimal execution problem in a continuous-time market model that considers market impact. We formulate the problem as a stochastic control problem and investigate properties of the corresponding value function. We find that right-continuity at the time origin is associated with the strength of market impact for large sales, otherwise the value function is continuous. Moreover, we show the semi-group property (Bellman principle) and characterise the value function as a viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation. We introduce some examples where the forms of the optimal strategies change completely, depending on the amount of the trader's security holdings and where optimal strategies in the Black-Scholes type market with nonlinear market impact are not block liquidation but gradual liquidation, even when the trader is risk-neutral.Comment: 36 pages, 8 figures, a modified version of the article "An optimal execution problem with market impact" in Finance and Stochastics (2014

The Tychonoff uniqueness theorem for the G-heat equation

In this paper, we obtain the Tychonoff uniqueness theorem for the G-heat equation