Non-minimality of corners in subriemannian geometry

We give a short solution to one of the main open problems in subriemannian geometry. Namely, we prove that length minimizers do not have corner-type singularities. With this result we solve Problem II of Agrachev's list, and provide the first general result toward the 30-year-old open problem of regularity of subriemannian geodesics.Comment: 11 pages, final versio

Outlier robust corner-preserving methods for reconstructing noisy images

The ability to remove a large amount of noise and the ability to preserve most structure are desirable properties of an image smoother. Unfortunately, they usually seem to be at odds with each other; one can only improve one property at the cost of the other. By combining M-smoothing and least-squares-trimming, the TM-smoother is introduced as a means to unify corner-preserving properties and outlier robustness. To identify edge- and corner-preserving properties, a new theory based on differential geometry is developed. Further, robustness concepts are transferred to image processing. In two examples, the TM-smoother outperforms other corner-preserving smoothers. A software package containing both the TM- and the M-smoother can be downloaded from the Internet.Comment: Published at http://dx.doi.org/10.1214/009053606000001109 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Majorana states in prismatic core-shell nanowires

We consider core-shell nanowires with conductive shell and insulating core, and with polygonal cross section. We investigate the implications of this geometry on Majorana states expected in the presence of proximity-induced superconductivity and an external magnetic field. A typical prismatic nanowire has a hexagonal profile, but square and triangular shapes can also be obtained. The low-energy states are localized at the corners of the cross section, i.e. along the prism edges, and are separated by a gap from higher energy states localized on the sides. The corner localization depends on the details of the shell geometry, i.e. thickness, diameter, and sharpness of the corners. We study systematically the low-energy spectrum of prismatic shells using numerical methods and derive the topological phase diagram as a function of magnetic field and chemical potential for triangular, square, and hexagonal geometries. A strong corner localization enhances the stability of Majorana modes to various perturbations, including the orbital effect of the magnetic field, whereas a weaker localization favorizes orbital effects and reduces the critical magnetic field. The prismatic geometry allows the Majorana zero-energy modes to be accompanied by low-energy states, which we call pseudo Majorana, and which converge to real Majoranas in the limit of small shell thickness. We include the Rashba spin-orbit coupling in a phenomenological manner, assuming a radial electric field across the shell.Comment: 14 pages, 16 figures, accepted for publication in Phys. Rev.

Ab initio Calculations of Multilayer Relaxations of Stepped Cu Surfaces

We present trends in the multilayer relaxations of several vicinals of Cu(100) and Cu(111) of varying terrace widths and geometry. The electronic structure calculations are based on density functional theory in the local density approximation with norm-conserving, non-local pseudopotentials in the mixed basis representation. While relaxations continue for several layers, the major effect concentrates near the step and corner atoms. On all surfaces the step atoms contract inwards, in agreement with experimental findings. Additionally, the corner atoms move outwards and the atoms in the adjacent chain undergo large inward relaxation. Correspondingly, the largest contraction (4%) is in the bond length between the step atom and its bulk nearest neighbor (BNN), while that between the corner atom and BNN is somewhat enlarged. The surface atoms also display changes in registry of upto 1.5%. Our results are in general in good agreement with LEED data including the controversial case of Cu(511). Subtle differences are found with results obtained from semi-empirical potentials.Comment: 21 pages and 3 figure

Second Order Topological Superconductivity in π\pi-Junction Rashba Layers

We consider a Josephson junction bilayer consisting of two tunnel-coupled two-dimensional electron gas layers with Rashba spin-orbit interaction, proximitized by a top and bottom $s$-wave superconductor with phase difference $\phi$ close to $\pi$. We show that, in the presence of a finite weak in-plane Zeeman field, the bilayer can be driven into a second order topological superconducting phase, hosting two Majorana corner states (MCSs). If $\phi=\pi$, in a rectangular geometry, these zero-energy bound states are located at two opposite corners determined by the direction of the Zeeman field. If the phase difference $\phi$ deviates from $\pi$ by a critical value, one of the two MCSs gets relocated to an adjacent corner. As the phase difference $\phi$ increases further, the system becomes trivially gapped. The obtained MCSs are robust against static and magnetic disorder. We propose two setups that could realize such a model: one is based on controlling $\phi$ by magnetic flux, the other involves an additional layer of randomly-oriented magnetic impurities responsible for the phase shift of $\pi$ in the proximity-induced superconducting pairing

Effect of port corner geometry on the internal performance of a rotating-vane-type thrust reverser

An investigation has been conducted in the static-test facility of the Langley 16-Foot Transonic Tunnel to determine the effects of reverser port geometry on the internal performance of a nonaxisymmetric rotating-vane-type thrust reverser. Thrust reverser vane positions representing a spoiled-trust (partially deployed) position and a full-reverse-thrust (fully deployed) position were tested with each port geometry variable. The effects of upstream port corner radius and wall angle on internal performance were determined. In addition, the effect of the length of a simulated cooling liner (blunt-base step) near the reverser port entrance was investigated; five different lengths were tested. All tests were conducted with no external flows, and nozzle pressure ratio was varied from 1.2 to 5.0

Conformal Mapping and Bound States in Bent Waveguides

Is it possible to trap a quantum particle in an open geometry? In this work we deal with the boundary value problem of the stationary Schroedinger (or Helmholtz) equation within a waveguide with straight segments and a rectangular bending. The problem can be reduced to a one dimensional matrix Schroedinger equation using two descriptions: oblique modes and conformal coordinates. We use a corner-corrected WKB formalism to find the energies of the one dimensional problem. It is shown that the presence of bound states is an effect due to the boundary alone, with no classical counterpart for this geometry. The conformal description proves to be simpler, as the coupling of transversal modes is not essential in this case.Comment: 16 pages, 10 figures. To appear in the Proceedings of the Symposium "Symmetries in Nature, in memoriam Marcos Moshinsky