A new construction of Lagrangians in the complex Euclidean plane in terms of planar curves

We introduce a new method to construct a large family of Lagrangian surfaces in complex Euclidean plane by means of two planar curves making use of their usual product as complex functions and integrating the Hermitian product of their position and tangent vectors. Among this family, we characterize minimal, constant mean curvature, Hamiltonian stationary, solitons for mean curvature flow and Willmore surfaces in terms of simple properties of the curvatures of the generating curves. As an application, we provide explicitly conformal parametrizations of known and new examples of these classes of Lagrangians in complex Euclidean plane.Comment: 15 pages, 5 figure

The application of gibberellic acid increases berry size of ‘Emperatriz’ seedless grape

Gibberellic acid (GA3) increases berry size of "Emperatriz" seedless grape, the response depending on the phenological stage of vine at treatment date and on the concentration applied. From berry fruit set to 21 days later, 80 mg/L GA3 increased commercial berry weight by 50%-90%, depending on the year, reaching similar size to that of "Aledo" seeded grape, used as comparison. This effect takes place through: a) a larger berry growth rate; b) an early glucose, fructose and sucrose uptake; c) an increase of absolute glucose and fructose content (mg/berry) of seedless berries up to similar values to those of seeded berries; and d) an increase of absolute berry water content but not of relative content to fresh weight, thus water potential and osmotic potential are not significantly modified by treatments. GA3 does not affect berry pericarp cell number but increases pericarp cell diameter

Approximated integrability of the Dicke model

A very approximate second integral of motion of the Dicke model is identified within a broad region above the ground state, and for a wide range of values of the external parameters. This second integral, obtained from a Born Oppenheimer approximation, classifies the whole regular part of the spectrum in bands labelled by its corresponding eigenvalues. Results obtained from this approximation are compared with exact numerical diagonalization for finite systems in the superradiant phase, obtaining a remarkable accord. The region of validity of our approach in the parameter space, which includes the resonant case, is unveiled. The energy range of validity goes from the ground state up to a certain upper energy where chaos sets in, and extends far beyond the range of applicability of a simple harmonic approximation around the minimal energy configuration. The upper energy validity limit increases for larger values of the coupling constant and the ratio between the level splitting and the frequency of the field. These results show that the Dicke model behaves like a two-degree of freedom integrable model for a wide range of energies and values of the external parameters.Comment: 6 pages, 3 figures. Second version with added text, references and some new figure

Integrable two-channel p_x+ip_y-wave superfluid model

We present a new two-channel integrable model describing a system of spinless fermions interacting through a p-wave Feshbach resonance. Unlike the BCS-BEC crossover of the s-wave case, the p-wave model has a third order quantum phase transition. The critical point coincides with the deconfinement of a single molecule within a BEC of bound dipolar molecules. The exact many-body wavefunction provides a unique perspective of the quantum critical region suggesting that the size of the condensate wavefunction, that diverges logarithmically with the chemical potential, could be used as an experimental indicator of the phase transition.Comment: 4 pages, 4 figure