Nonholonomic LR systems as Generalized Chaplygin systems with an Invariant Measure and Geodesic Flows on Homogeneous Spaces

We consider a class of dynamical systems on a Lie group $G$ with a left-invariant metric and right-invariant nonholonomic constraints (so called LR systems) and show that, under a generic condition on the constraints, such systems can be regarded as generalized Chaplygin systems on the principle bundle $G \to Q=G/H$, $H$ being a Lie subgroup. In contrast to generic Chaplygin systems, the reductions of our LR systems onto the homogeneous space $Q$ always possess an invariant measure. We study the case $G=SO(n)$, when LR systems are multidimensional generalizations of the Veselova problem of a nonholonomic rigid body motion, which admit a reduction to systems with an invariant measure on the (co)tangent bundle of Stiefel varieties $V(k,n)$ as the corresponding homogeneous spaces. For $k=1$ and a special choice of the left-invariant metric on SO(n), we prove that under a change of time, the reduced system becomes an integrable Hamiltonian system describing a geodesic flow on the unit sphere $S^{n-1}$. This provides a first example of a nonholonomic system with more than two degrees of freedom for which the celebrated Chaplygin reducibility theorem is applicable. In this case we also explicitly reconstruct the motion on the group SO(n).Comment: 39 pages, the proof of Lemma 4.3 and some references are added, to appear in Journal of Nonlinear Scienc

In-plane object detection : detection algorithms and visibility problems

A large number of devices today incorporate some form of detection of objects and people in a given environment. Various detection technologies have been developed over the years, as a response to many different demands. The devices such as video surveillance systems, scanners, touch screens and various systems for tracking people and objects in space, detect objects using camera videos and/or measurements gathered by sensors. To enable simultaneous detection of multiple objects on table-top interactive devices designed to support games that combine the social attractiveness of traditional board games with the interactivity of computer games, an in-plane detection technology that uses LEDs and sensors was developed by Philips. The presence of objects on the table results in blocking light emitted by the LEDs for some of the sensors. This information can be used to determine the position and shape of objects such as game pieces or fingers on the table. If the detection process is performed fast enough, then moving objects can be tracked, for instance, to recognize gestures made by fingers. This detection technique gives rise to many interesting geometric problems, such as developing efficient detection algorithms. In addition, due to occlusion created by the multiple objects placed on the table, some visibility problems can occur in the process of detecting objects. We present detection algorithms that use the sensor data as an input and provide an approximation on the geometry of objects as an output. We discuss the advantages and disadvantages of the presented algorithms and analyze their worst-case time complexity as a function of the number of LEDs and sensors. In addition, the maximum level of the accuracy of detecting circular objects that can be achieved has been investigated. To investigate this maximum level of accuracy we assume infinitely many LEDs and sensors in a frame surrounding the objects. We present and discuss a worst-case optimal algorithm that determines the output that a detection algorithm would provide in this case. Several visibility problems have been explored that relate to occlusion, an intrinsic shortcoming of the detection technique. Among many visibility problems that can be identified, the focus was on five problems related to either falsely detecting a non-existing circular object or detecting multiple objects as one. These problems occur when multiple objects positioned in the detection area block all of the lines of sight between LEDs and sensors that cross some area that is not occupied by an object. In this thesis, we focused on exploring the worst-case scenarios, in other words, finding the minimum number of identical circular objects that can cause one such visibility problem to occur in relation to the distance between the objects. We have proved that this number is quadratic in the minimum mutual distance between the objects. This result can be used in practice, for example, to adapt the layout of game boards such that these visibility problems can be avoided

Optically pumped intersublevel midinfrared lasers based on InAs-GaAs quantum dots

We propose an optically pumped laser based on intersublevel transitions in InAs-GaAs pyramidal self-Assembled quantum dots. A theoretical rate equations model of the laser is given in order to predict the dependence of the gain on pumping flux and temperature. The energy levels and wave functions were calculated using the 8-band k . p method where the symmetry of the pyramid was exploited to reduce the computational complexity. Carrier dynamics in the laser were modeled by taking both electron-longitudinal optical phonon and electron-longitudinal acoustic phonon interactions into account. The proposed laser emits at 14.6 μm with a gain of g ≈ 570 cm(-1) at the pumping flux Φ= 10(24) cm(-2) s(-1) and a temperature of T = 77 K. By varying the size of the investigated dots, laser emission in the spectral range 13-21 μm is predicted. In comparison to optically pumped lasers based on quantum wells, an advantage of the proposed type of laser is a lower pumping flux, due to the longer carrier lifetime in quantum dots, and also that both surface and edge emission are possible. The appropriate waveguide and cavity designs are presented, and by comparing the calculated values of the gain with the estimated losses, lasing is predicted even at room temperature for all the quantum dots investigated

Asymptotic bounds on minimum number of disks required to hide a disk

We consider the problem of blocking all rays emanating from a closed unit disk with a minimum number of closed unit disks in the two-dimensional space, where the minimum distance from a disk to any other disk is given. We study the asymptotic behavior of the minimum number of disks as the minimum mutual distance approaches infinity. Using a regular ordering of disks on concentric circular rings we derive an upper bound and prove that the minimum number of disks required for blocking is quadratic in the minimum distance between the disks

The Subaru Coronagraphic Extreme Adaptive Optics System: Enabling High-Contrast Imaging on Solar-System Scales

The Subaru Coronagraphic Extreme Adaptive Optics (SCExAO) instrument is a multipurpose high-contrast imaging platform designed for the discovery and detailed characterization of exoplanetary systems and serves as a testbed for high-contrast imaging technologies for ELTs. It is a multiband instrument which makes use of light from 600 to 2500 nm, allowing for coronagraphic direct exoplanet imaging of the inner 3λ/D from the stellar host. Wavefront sensing and control are key to the operation of SCExAO. A partial correction of low-order modes is provided by Subaru's facility adaptive optics system with the final correction, including high-order modes, implemented downstream by a combination of a visible pyramid wavefront sensor and a 2000-element deformable mirror. The well-corrected NIR (y-K bands) wavefronts can then be injected into any of the available coronagraphs, including but not limited to the phase-induced amplitude apodization and the vector vortex coronagraphs, both of which offer an inner working angle as low as 1λ/D. Noncommon path, low-order aberrations are sensed with a coronagraphic low-order wavefront sensor in the infrared (IR). Low noise, high frame rate NIR detectors allow for active speckle nulling and coherent differential imaging, while the HAWAII 2RG detector in the HiCIAO imager and/or the CHARIS integral field spectrograph (from mid-2016) can take deeper exposures and/or perform angular, spectral, and polarimetric differential imaging. Science in the visible is provided by two interferometric modules: VAMPIRES and FIRST, which enable subdiffraction limited imaging in the visible region with polarimetric and spectroscopic capabilities respectively. We describe the instrument in detail and present preliminary results both on-sky and in the laboratory

Optically pumped terahertz laser based on intersubband transitions in a GaN/AlGaN double quantum well

A design for a GaN/AlGaN optically pumped terahertz laser emitting at 34 µm (ΔE~36 meV) is presented. This laser uses a simple three-level scheme where the depopulation of the lower laser level is achieved via resonant longitudinal-optical-phonon emission. The quasibound energies and associated wave functions are calculated with the intrinsic electric field induced by the piezoelectric and the spontaneous polarizations. The structures based on a double quantum well were simulated and the output characteristics extracted using a fully self-consistent rate equation model with all relevant scattering processes included. Both electron-longitudinal-optical phonon and electron-acoustic-phonon interactions were taken into account. The carrier distribution in subbands was assumed to be Fermi–Dirac-like, with electron temperature equal to the lattice temperature, but with different Fermi levels for each subband. A population inversion of 12% for a pumping flux Φ=10(27) cm(–2) s(–1) at room temperature was calculated for the optimized structure. By comparing the calculated modal gain and estimated waveguide and mirror losses the feasibility of laser action up to room temperature is predicted

Nanosecond-timescale spin transfer using individual electrons in a quadruple-quantum-dot device

The ability to coherently transport electron-spin states between different sites of gate-defined semiconductor quantum dots is an essential ingredient for a quantum-dot-based quantum computer. Previous shuttles using electrostatic gating were too slow to move an electron within the spin dephasing time across an array. Here we report a nanosecond-timescale spin transfer of individual electrons across a quadruple-quantum-dot device. Utilizing enhanced relaxation rates at a so-called `hot spot', we can upper bound the shuttle time to at most 150 ns. While actual shuttle times are likely shorter, 150 ns is already fast enough to preserve spin coherence in e.g. silicon based quantum dots. This work therefore realizes an important prerequisite for coherent spin transfer in quantum dot arrays.Comment: 7 pages including 2 pages of supplementary materia