On the Cardinality of Positively Linearly Independent Sets

Positive bases, which play a key role in understanding derivative free optimization methods that use a direct search framework, are positive spanning sets that are positively linearly independent. The cardinality of a positive basis in $\R^n$ has been established to be between $n+1$ and $2n$ (with both extremes existing). The lower bound is immediate from being a positive spanning set, while the upper bound uses {\em both} positive spanning and positively linearly independent. In this note, we provide details proving that a positively linearly independent set in $\R^n$ for $n \in \{1, 2\}$ has at most $2n$ elements, but a positively linearly independent set in $\R^n$ for $n\geq 3$ can have an arbitrary number of elements

Dynamics of a tight-binding ring threaded by time-periodic magnetic flux

We analytically study the effects of periodically alternating magnetic fields on the dynamics of a tight-binding ring. It is shown that an arbitrary quantum state can be frozen coherently at will by the very frequent square-wave field as well as the monochromatic-wave field when the corresponding optimal amplitudes are taken. Numerical simulations show that the average fidelity depends on not only the system parameters, but also the features of the quantum state. Moreover, taking the initial zero-momentum Gaussian wave packets as examples, we show the dependence of the threshold frequency on the width of the packet for the given average fidelities. These observations provide a means to perform the quantum state engineering.Comment: 8 pages, 9 figure

1.55 µm AlGaInAs/InP sampled grating laser diodes for mode-locking at THz frequencies

We report mode locking in lasers integrated with semiconductor optical amplifiers, using either conventional or phase shifted sampled grating distributed Bragg reflectors(DBRs). For a conventional sampled grating with a continuous grating coupling coefficient of ~80 cm-1, mode-locking was observed at a fundamental frequency of 628 GHz and second harmonic of 1.20 THz. The peak output power was up to 142 mW. In the phase shifted sampled grating design, the grating is present along the entire length of the reflector with π-phase shift steps within each sampled section. The effective coupling coefficient is therefore increased substantially. Although the continuous grating coupling coefficient for the phase shifted gratings was reduced to ~23 cm-1 because of a different fabrication technology, the lasers demonstrated mode locking at fundamental repetition frequencies of 620 GHz and 1 THz, with a much lower level of amplified spontaneous emission seen in the output spectra than from conventional sampled grating devices. Although high pulse reproducibility and controllability over a wide operation range was seen for both types of grating, the π-phase-shifted gratings already demonstrate fundamental mode-locking to 1 THz. The integrated semiconductor optical amplifier makes sampled grating DBR lasers ideal pump sources for generating THz signals through photomixing

VISHNU hybrid model for viscous QCD matter at RHIC and LHC energies

In this proceeding, we briefly describe the viscous hydrodynamics + hadron cascade hybrid model VISHNU for relativistic heavy ion collisions and report the current status on extracting the QGP viscosity from elliptic flow data.Comment: 4 pages, 1 figure, the proceedings of 7th International Workshop on Critical Point and Onset of Deconfinement, Wuhan, China, Nov. 7-11, 201

THz Repetition Frequency Mode-Locked Laser Using Novel Sampled Gratings

Conventional sampled grating distributed-Bragg-gratings (C-SGDBRs) are widely used in tunable DBR lasers [1], and more recently have been used to precisely control the wavelength spacing in arrays of DBR lasers for use in WDM systems [2], and as the reflectors in THz repetition frequency (Fr) semiconductor mode locked lasers (SMLLs) [3]. However, the effective coupling coefficient, κ, of a C-SGDBR (Fig. 1(a)) is necessarily reduced substantially from that of a uniform grating because much of the sampled grating period has no grating. Here, for the first time, we apply a combination of π-phase shifted gratings, previously demonstrated in fiber lasers [4], with the C-SGDBR technique to THz repetition frequency SMLLs. Using a single electron beam lithography (EBL) step we have demonstrated a 620 GHz side-wall SGDBR MLL with an increased effective κ