SL(2,R)-geometric phase space and (2+2)-dimensions

We propose an alternative geometric mathematical structure for arbitrary phase space. The main guide in our approach is the hidden SL(2,R)-symmetry which acts on the phase space changing coordinates by momenta and vice versa. We show that the SL(2,R)-symmetry is implicit in any symplectic structure. We also prove that in any sensible physical theory based on the SL(2,R)-symmetry the signature of the flat target "spacetime" must be associated with either one-time and one-space or at least two-time and two-space coordinates. We discuss the consequences as well as possible applications of our approach on different physical scenarios.Comment: 17 pages, no figure

Superintegrability of the Fock-Darwin system

The Fock-Darwin system is analysed from the point of view of its symmetry properties in the quantum and classical frameworks. The quantum Fock-Darwin system is known to have two sets of ladder operators, a fact which guarantees its solvability. We show that for rational values of the quotient of two relevant frequencies, this system is superintegrable, the quantum symmetries being responsible for the degeneracy of the energy levels. These symmetries are of higher order and close a polynomial algebra. In the classical case, the ladder operators are replaced by ladder functions and the symmetries by constants of motion. We also prove that the rational classical system is superintegrable and its trajectories are closed. The constants of motion are also generators of symmetry transformations in the phase space that have been integrated for some special cases. These transformations connect different trajectories with the same energy. The coherent states of the quantum superintegrable system are found and they reproduce the closed trajectories of the classical one.Comment: 21 pages,16 figure

Towards an Ashtekar formalism in eight dimensions

We investigate the possibility of extending the Ashtekar theory to eight dimensions. Our approach relies on two notions: the octonionic structure and the MacDowell-Mansouri formalism generalized to a spacetime of signature 1+7. The key mathematical tool for our construction is the self-dual (antiself-dual) four-rank fully antisymmetric octonionic tensor. Our results may be of particular interest in connection with a possible formulation of M-theory via matroid theory.Comment: 15 pages, Latex, minor changes, to appear in Class. Quantum Gra

Assessing the Formation Scenarios for the Double Nucleus of M31 Using Two-Dimensional Image Decomposition

The double nucleus geometry of M31 is currently best explained by the eccentric disk hypothesis of Tremaine, but whether the eccentric disk resulted from the tidal disruption of an inbounding star cluster by a nuclear black hole, or by an m=1 perturbation of a native nuclear disk, remains debatable. I perform detailed 2-D decomposition of the M31 double nucleus in the Hubble Space Telescope V-band to study the bulge structure and to address competing formation scenarios of the eccentric disk. I deblend the double nucleus (P1 and P2) and the bulge simultaneously using five Sersic and one Nuker components. P1 and P2 appear to be embedded inside an intermediate component (r_e=3.2") that is nearly spherical (q=0.97+/-m0.02), while the main galaxy bulge is more elliptical (q=0.81+/-0.01). The spherical bulge mass of 2.8x10^7 M_sol is comparable to the supermassive black hole mass (3x10^7 M_sol). In the 2-D decomposition, the bulge is consistent with being centered near the UV peak of P2, but the exact position is difficult to pinpoint because of dust in the bulge. P1 and P2 are comparable in mass. Within a radius r=1\arcsec of P2, the relative mass fraction of the nuclear components is M_BH:M_bulge:P1: P2 = 4.3:1.2:1:0.7, assuming the luminous components have a common mass-to-light ratio of 5.7. The eccentric disk as a whole (P1+P2) is massive, M ~ 2.1x10^7 M_sol, comparable to the black hole and the local bulge mass. As such, the eccentric disk could not have been formed entirely out of stars that were stripped from an inbounding star cluster. Hence, the more favored scenario is that of a disk formed in situ by an m=1 perturbation, caused possibly by the passing of a giant molecular cloud, or the passing/accretion of a small globular cluster.Comment: 19 pages, 8 figures. AJ accepted. For the version of this paper with high resolution figures, go to: http://zwicky.as.arizona.edu/~cyp/work/m31.ps.g

Influence of the Ground-State Topology on the Domain-Wall Energy in the Edwards-Anderson +/- J Spin Glass Model

We study the phase stability of the Edwards-Anderson spin-glass model by analyzing the domain-wall energy. For the bimodal distribution of bonds, a topological analysis of the ground state allows us to separate the system into two regions: the backbone and its environment. We find that the distributions of domain-wall energies are very different in these two regions for the three dimensional (3D) case. Although the backbone turns out to have a very high phase stability, the combined effect of these excitations and correlations produces the low global stability displayed by the system as a whole. On the other hand, in two dimensions (2D) we find that the surface of the excitations avoids the backbone. Our results confirm that a narrow connection exists between the phase stability of the system and the internal structure of the ground-state. In addition, for both 3D and 2D we are able to obtain the fractal dimension of the domain wall by direct means.Comment: 4 pages, 3 figures. Accepted for publication in Rapid Communications of Phys. Rev.

Derivation of a multilayer approach to model suspended sediment transport: application to hyperpycnal and hypopycnal plumes

We propose a multi-layer approach to simulate hyperpycnal and hypopycnal plumes in flows with free surface. The model allows to compute the vertical profile of the horizontal and the vertical components of the velocity of the fluid flow. The model can describe as well the vertical profile of the sediment concentration and the velocity components of each one of the sediment species that form the turbidity current. To do so, it takes into account the settling velocity of the particles and their interaction with the fluid. This allows to better describe the phenomena than a single layer approach. It is in better agreement with the physics of the problem and gives promising results. The numerical simulation is carried out by rewriting the multi-layer approach in a compact formulation, which corresponds to a system with non-conservative products, and using path-conservative numerical scheme. Numerical results are presented in order to show the potential of the model

2D granular flows with the μ(I)\mu(I) rheology and side walls friction: a well balanced multilayer discretization

We present here numerical modelling of granular flows with the $\mu(I)$ rheology in confined channels. The contribution is twofold: (i) a model to approximate the Navier-Stokes equations with the $\mu(I)$ rheology through an asymptotic analysis. Under the hypothesis of a one-dimensional flow, this model takes into account side walls friction; (ii) a multilayer discretization following Fern\'andez-Nieto et al. (J. Fluid Mech., vol. 798, 2016, pp. 643-681). In this new numerical scheme, we propose an appropriate treatment of the rheological terms through a hydrostatic reconstruction which allows this scheme to be well-balanced and therefore to deal with dry areas. Based on academic tests, we first evaluate the influence of the width of the channel on the normal profiles of the downslope velocity thanks to the multilayer approach that is intrinsically able to describe changes from Bagnold to S-shaped (and vice versa) velocity profiles. We also check the well balance property of the proposed numerical scheme. We show that approximating side walls friction using single-layer models may lead to strong errors. Secondly, we compare the numerical results with experimental data on granular collapses. We show that the proposed scheme allows us to qualitatively reproduce the deposit in the case of a rigid bed (i. e. dry area) and that the error made by replacing the dry area by a small layer of material may be large if this layer is not thin enough. The proposed model is also able to reproduce the time evolution of the free surface and of the flow/no-flow interface. In addition, it reproduces the effect of erosion for granular flows over initially static material lying on the bed. This is possible when using a variable friction coefficient $\mu(I)$ but not with a constant friction coefficient