Secants of Lagrangian Grassmannians

We study the dimensions of secant varieties of the Grassmannian of Lagrangian subspaces in a symplectic vector space. We calculate these dimensions for third and fourth secant varieties. Our result is obtained by providing a normal form for four general points on such a Grassmannian and by explicitly calculating the tangent spaces at these four points

On the dimensions of secant varieties of Segre-Veronese varieties

This paper explores the dimensions of higher secant varieties to Segre-Veronese varieties. The main goal of this paper is to introduce two different inductive techniques. These techniques enable one to reduce the computation of the dimension of the secant variety in a high dimensional case to the computation of the dimensions of secant varieties in low dimensional cases. As an application of these inductive approaches, we will prove non-defectivity of secant varieties of certain two-factor Segre-Veronese varieties. We also use these methods to give a complete classification of defective s-th Segre-Veronese varieties for small s. In the final section, we propose a conjecture about defective two-factor Segre-Veronese varieties.Comment: Revised version. To appear in Annali di Matematica Pura e Applicat

Dissociation and Decay of Ultra-cold Sodium Molecules

The dissociation of ultracold molecules is studied by ramping an external magnetic field through a Feshbach resonance. The observed dissociation energy shows non-linear dependence on the ramp speed and directly yields the strength of the atom-molecule coupling. In addition, inelastic molecule-molecule and molecule-atom collisions are characterized

Formation of Quantum-Degenerate Sodium Molecules

Ultra-cold sodium molecules were produced from an atomic Bose-Einstein condensate by ramping an applied magnetic field across a Feshbach resonance. More than $10^5$ molecules were generated with a conversion efficiency of $\sim$4%. Using laser light resonant with an atomic transition, the remaining atoms could be selectively removed, preventing fast collisional relaxation of the molecules. Time-of-flight analysis of the pure molecular sample yielded an instantaneous phase-space density greater than 20.Comment: 5 pages, 4 figures (final published version

Giant Vortex Lattice Deformations in Rapidly Rotating Bose-Einstein Condensates

We have performed numerical simulations of giant vortex structures in rapidly rotating Bose-Einstein condensates within the Gross-Pitaevskii formalism. We reproduce the qualitative features, such as oscillation of the giant vortex core area, formation of toroidal density hole, and the precession of giant vortices, observed in the recent experiment [Engels \emph{et.al.}, Phys. Rev. Lett. {\bf 90}, 170405 (2003)]. We provide a mechanism which quantitatively explains the observed core oscillation phenomenon. We demonstrate the clear distinction between the mechanism of atom removal and a repulsive pinning potential in creating giant vortices. In addition, we have been able to simulate the transverse Tkachenko vortex lattice vibrations.Comment: 5 pages, 6 figures; revised description of core oscillation, new subfigur

Vortex phase diagram in rotating two-component Bose-Einstein condensates

We investigate the structure of vortex states in rotating two-component Bose-Einstein condensates with equal intracomponent but varying intercomponent coupling constants. A phase diagram in the intercomponent-coupling versus rotation-frequency plane reveals rich equilibrium structures of vortex states. As the ratio of intercomponent to intracomponent couplings increases, the interlocked vortex lattices undergo phase transitions from triangular to square, to double-core lattices, and eventually develop interwoven "serpentine" vortex sheets with each component made up of chains of singly quantized vortices.Comment: 4 pages, 4 figures, revtex

Coherent Molecular Optics using Sodium Dimers

Coherent molecular optics is performed using two-photon Bragg scattering. Molecules were produced by sweeping an atomic Bose-Einstein condensate through a Feshbach resonance. The spectral width of the molecular Bragg resonance corresponded to an instantaneous temperature of 20 nK, indicating that atomic coherence was transferred directly to the molecules. An autocorrelating interference technique was used to observe the quadratic spatial dependence of the phase of an expanding molecular cloud. Finally, atoms initially prepared in two momentum states were observed to cross-pair with one another, forming molecules in a third momentum state. This process is analogous to sum-frequency generation in optics

Evaluation of Flashover Voltage Levels of Contaminated Hydrophobic Polymer Insulators Using Regression Trees, Neural Networks, and Adaptive Neuro-Fuzzy

Polluted insulators at high voltages has acquired considerable importance with the rise of voltage transmission lines. The contamination may lead to flashover voltage. As a result, flashover voltage could lead to service outage and affects negatively the reliability of the power system. This paper presents a dynamic model of ac 50Hz flashover voltages of polluted hydrophobic polymer insulators. The models are constructed using the regression tree method, artificial neural network (ANN), and adaptive neuro-fuzzy (ANFIS). For this purpose, more than 2000 different experimental testing conditions were used to generate a training set. The study of the ac flashover voltages depends on silicone rubber (SiR) percentage content in ethylene propylene diene monomer (EPDM) rubber. Besides, water conductivity (μS/cm), number of droplets on the surface, and volume of water droplet (ml) are considered. The regression tree model is obtained and the performance of the proposed system with other intelligence methods is compa ed. It can be concluded that the performance of the least squares regression tree model outperforms the other intelligence methods, which gives the proposed model better generalization ability

Tkachenko modes of vortex lattices in rapidly rotating Bose-Einstein condensates

We calculate the in-plane modes of the vortex lattice in a rotating Bose condensate from the Thomas-Fermi to the mean-field quantum Hall regimes. The Tkachenko mode frequency goes from linear in the wavevector, $k$, for lattice rotational velocities, $\Omega$, much smaller than the lowest sound wave frequency in a finite system, to quadratic in $k$ in the opposite limit. The system also supports an inertial mode of frequency $\ge 2\Omega$. The calculated frequencies are in good agreement with recent observations of Tkachenko modes at JILA, and provide evidence for the decrease in the shear modulus of the vortex lattice at rapid rotation.Comment: 4 pages, 2 figure