The family of quaternionic quasi-unitary Lie algebras and their central extensions

The family of quaternionic quasi-unitary (or quaternionic unitary Cayley--Klein algebras) is described in a unified setting. This family includes the simple algebras sp(N+1) and sp(p,q) in the Cartan series C_{N+1}, as well as many non-semisimple real Lie algebras which can be obtained from these simple algebras by particular contractions. The algebras in this family are realized here in relation with the groups of isometries of quaternionic hermitian spaces of constant holomorphic curvature. This common framework allows to perform the study of many properties for all these Lie algebras simultaneously. In this paper the central extensions for all quasi-simple Lie algebras of the quaternionic unitary Cayley--Klein family are completely determined in arbitrary dimension. It is shown that the second cohomology group is trivial for any Lie algebra of this family no matter of its dimension.Comment: 17 pages, LaTe

Low-power optical beam steering by microelectromechanical waveguide gratings

Optical beam steering is key for optical communications, laser mapping (LIDAR), and medical imaging. For these applications, integrated photonics is an enabling technology that can provide miniaturized, lighter, lower cost, and more power efficient systems. However, common integrated photonic devices are too power demanding. Here, we experimentally demonstrate, for the first time, beam steering by microelectromechanical (MEMS) actuation of a suspended silicon photonic waveguide grating. Our device shows up to 5.6{\deg} beam steering with 20 V actuation and a power consumption below the $\mu$W level, i.e. more than 5 orders of magnitude lower power consumption than previous thermo-optic tuning methods. The novel combination of MEMS with integrated photonics presented in this work lays ground for the next generation of power-efficient optical beam steering systems

Central extensions of the families of quasi-unitary Lie algebras

The most general possible central extensions of two whole families of Lie algebras, which can be obtained by contracting the special pseudo-unitary algebras su(p,q) of the Cartan series A_l and the pseudo-unitary algebras u(p,q), are completely determined and classified for arbitrary p,q. In addition to the su(p,q) and u({p,q}) algebras, whose second cohomology group is well known to be trivial, each family includes many non-semisimple algebras; their central extensions, which are explicitly given, can be classified into three types as far as their properties under contraction are involved. A closed expression for the dimension of the second cohomology group of any member of these families of algebras is given.Comment: 23 pages. Latex2e fil

A Bayesian approach to filter design: detection of compact sources

We consider filters for the detection and extraction of compact sources on a background. We make a one-dimensional treatment (though a generalization to two or more dimensions is possible) assuming that the sources have a Gaussian profile whereas the background is modeled by an homogeneous and isotropic Gaussian random field, characterized by a scale-free power spectrum. Local peak detection is used after filtering. Then, a Bayesian Generalized Neyman-Pearson test is used to define the region of acceptance that includes not only the amplification but also the curvature of the sources and the a priori probability distribution function of the sources. We search for an optimal filter between a family of Matched-type filters (MTF) modifying the filtering scale such that it gives the maximum number of real detections once fixed the number density of spurious sources. We have performed numerical simulations to test theoretical ideas.Comment: 10 pages, 2 figures. SPIE Proceedings "Electronic Imaging II", San Jose, CA. January 200

The estimation of the SZ effects with unbiased multifilters

In this work we study the performance of linear multifilters for the estimation of the amplitudes of the thermal and kinematic Sunyaev-Zel'dovich effects. We show that when both effects are present, estimation of these effects with standard matched multifilters is intrinsically biased. This bias is due to the fact that both signals have basically the same spatial profile. We find a new family of multifilters related to the matched multifilters that cancel this systematic bias, hence we call them Unbiased Matched Multifilters. We test the unbiased matched multifilters and compare them with the standard matched multifilters using simulations that reproduce the future Planck mission's observations. We find that in the case of the standard matched multifilters the systematic bias in the estimation of the kinematic Sunyaev-Zel'dovich effect can be very large, even greater than the statistical error bars. Unbiased matched multifilters cancel effectively this kind of bias. In concordance with other works in the literature, our results indicate that the sensitivity and resolution of Planck will not be enough to give reliable estimations of the kinematic Sunyaev-Zel'dovich of individual clusters. However, since the estimation with the unbiased matched multifilters is not intrinsically biased, it can be possible to use them to statistically study peculiar velocities in large scales using large sets of clusters.Comment: 12 pages, 6 figures, submitted to MNRA

On the regularity of the covariance matrix of a discretized scalar field on the sphere

We present a comprehensive study of the regularity of the covariance matrix of a discretized field on the sphere. In a particular situation, the rank of the matrix depends on the number of pixels, the number of spherical harmonics, the symmetries of the pixelization scheme and the presence of a mask. Taking into account the above mentioned components, we provide analytical expressions that constrain the rank of the matrix. They are obtained by expanding the determinant of the covariance matrix as a sum of determinants of matrices made up of spherical harmonics. We investigate these constraints for five different pixelizations that have been used in the context of Cosmic Microwave Background (CMB) data analysis: Cube, Icosahedron, Igloo, GLESP and HEALPix, finding that, at least in the considered cases, the HEALPix pixelization tends to provide a covariance matrix with a rank closer to the maximum expected theoretical value than the other pixelizations. The effect of the propagation of numerical errors in the regularity of the covariance matrix is also studied for different computational precisions, as well as the effect of adding a certain level of noise in order to regularize the matrix. In addition, we investigate the application of the previous results to a particular example that requires the inversion of the covariance matrix: the estimation of the CMB temperature power spectrum through the Quadratic Maximum Likelihood algorithm. Finally, some general considerations in order to achieve a regular covariance matrix are also presented.Comment: 36 pages, 12 figures; minor changes in the text, matches published versio

Universal integrals for superintegrable systems on N-dimensional spaces of constant curvature

An infinite family of classical superintegrable Hamiltonians defined on the N-dimensional spherical, Euclidean and hyperbolic spaces are shown to have a common set of (2N-3) functionally independent constants of the motion. Among them, two different subsets of N integrals in involution (including the Hamiltonian) can always be explicitly identified. As particular cases, we recover in a straightforward way most of the superintegrability properties of the Smorodinsky-Winternitz and generalized Kepler-Coulomb systems on spaces of constant curvature and we introduce as well new classes of (quasi-maximally) superintegrable potentials on these spaces. Results here presented are a consequence of the sl(2) Poisson coalgebra symmetry of all the Hamiltonians, together with an appropriate use of the phase spaces associated to Poincare and Beltrami coordinates.Comment: 12 page

Integrable potentials on spaces with curvature from quantum groups

A family of classical integrable systems defined on a deformation of the two-dimensional sphere, hyperbolic and (anti-)de Sitter spaces is constructed through Hamiltonians defined on the non-standard quantum deformation of a sl(2) Poisson coalgebra. All these spaces have a non-constant curvature that depends on the deformation parameter z. As particular cases, the analogues of the harmonic oscillator and Kepler--Coulomb potentials on such spaces are proposed. Another deformed Hamiltonian is also shown to provide superintegrable systems on the usual sphere, hyperbolic and (anti-)de Sitter spaces with a constant curvature that exactly coincides with z. According to each specific space, the resulting potential is interpreted as the superposition of a central harmonic oscillator with either two more oscillators or centrifugal barriers. The non-deformed limit z=0 of all these Hamiltonians can then be regarded as the zero-curvature limit (contraction) which leads to the corresponding (super)integrable systems on the flat Euclidean and Minkowskian spaces.Comment: 19 pages, 1 figure. Two references adde