Quantum chaos with non-periodic, complex orbits in the Resonant Tunneling Diode

We show that a special type of orbits, which are non-periodic and complex, 'Saddle Orbits' (SOs) describe accurately the quantal and experimental current oscillations in the Resonant Tunneling Diode in tilted fields. This is the first demonstration that one needs to abandon the periodic orbits (POs) paradigm in favour of this new and peculiar type of orbit. The SOs solve the puzzle of broad regions of experimental oscillations where we find no real or complex PO that can explain the data. The SOs succeeds in regimes involving several non-isolated POs, where PO formulas fail.Comment: 3 pages, 5 figure

Black holes and the double copy

Recently, a perturbative duality between gauge and gravity theories (the double copy) has been discovered, that is believed to hold to all loop orders. In this paper, we examine the relationship between classical solutions of non-Abelian gauge theory and gravity. We propose a general class of gauge theory solutions that double copy to gravity, namely those involving stationary Kerr-Schild metrics. The Schwarzschild and Kerr black holes (plus their higher-dimensional equivalents) emerge as special cases. We also discuss plane wave solutions. Furthermore, a recently examined double copy between the self-dual sectors of Yang-Mills theory and gravity can be reinterpreted using a momentum-space generalisation of the Kerr-Schild framework.Comment: 22 pages; typos corrected and references adde

Convex Optimization Methods for Dimension Reduction and Coefficient Estimation in Multivariate Linear Regression

In this paper, we study convex optimization methods for computing the trace norm regularized least squares estimate in multivariate linear regression. The so-called factor estimation and selection (FES) method, recently proposed by Yuan et al. [22], conducts parameter estimation and factor selection simultaneously and have been shown to enjoy nice properties in both large and finite samples. To compute the estimates, however, can be very challenging in practice because of the high dimensionality and the trace norm constraint. In this paper, we explore a variant of Nesterov's smooth method [20] and interior point methods for computing the penalized least squares estimate. The performance of these methods is then compared using a set of randomly generated instances. We show that the variant of Nesterov's smooth method [20] generally outperforms the interior point method implemented in SDPT3 version 4.0 (beta) [19] substantially . Moreover, the former method is much more memory efficient.Comment: 27 page

Periodic Orbit theory for Resonant Tunneling Diodes : comparison with quantum and experimental results

We investigate whether the quantal and experimental amplitudes of current oscillations of Resonant Tunneling Diodes in tilted fields are obtainable from Periodic Orbit (PO) theories by considering recently proposed PO approaches. We show, for the first time, that accurate amplitudes and frequency shifts for the current oscillations (typically to within a few %) can be obtained from a simple analytical formula both in the stable (torus-quantization) limit and the unstable regimes of the experiments which are dominated by isolated PO's. But we find that the PO approach does not describe quantitatively the dynamically interesting intermediate experimental regimes which appear to be dominated by contributions from complex orbits and multiple non-isolated PO's. We conclude that these regimes will not easily be described by the usual PO approach, even with simple normal forms.Comment: 5 pages, 4 figure

Semiclassical description of resonant tunneling

We derive a semiclassical formula for the tunneling current of electrons trapped in a potential well which can tunnel into and across a wide quantum well. The calculations idealize an experimental situation where a strong magnetic field tilted with respect to an electric field is used. The resulting semiclassical expression is written as the sum over special periodic orbits which hit both walls of the quantum well and are perpendicular to the first wall.Comment: LaTeX, 8 page

What Determines the Success of a Geographical Indication? A Price-based Meta-Analysis for GIs In Food Products

Food Consumption/Nutrition/Food Safety,