Discretization of Fractional Differential Equations by a Piecewise Constant Approximation

There has recently been considerable interest in using a nonstandard piecewise approximation to formulate fractional order differential equations as difference equations that describe the same dynamical behaviour and are more amenable to a dynamical systems analysis. Unfortunately, due to mistakes in the fundamental papers, the difference equations formulated through this process do not capture the dynamics of the fractional order equations. We show that the correct application of this nonstandard piecewise approximation leads to a one parameter family of fractional order differential equations that converges to the original equation as the parameter tends to zero. A closed formed solution exists for each member of this family and leads to the formulation of a difference equation that is of increasing order as time steps are taken. Whilst this does not lead to a simplified dynamical analysis it does lead to a numerical method for solving the fractional order differential equation. The method is shown to be equivalent to a quadrature based method, despite the fact that it has not been derived from a quadrature. The method can be implemented with non-uniform time steps. An example is provided showing that the difference equation can correctly capture the dynamics of the underlying fractional differential equation

Supersymmetry, quark confinement and the harmonic oscillator

We study some quantum systems described by noncanonical commutation relations formally expressed as [q,p]=ihbar(I + chi H), where H is the associated (harmonic oscillator-like) Hamiltonian of the system, and chi is a Hermitian (constant) operator, i.e. [H,chi]=0 . In passing, we also consider a simple (chi=0 canonical) model, in the framework of a relativistic Klein-Gordon-like wave equation.Comment: To be published in Journal of Physics A: Mathematical and Theoretical (2007

Discovery of a 66 mas Ultracool Binary with Laser Guide Star Adaptive Optics

We present the discovery of 2MASS J21321145+1341584AB as a closely separated (0.066") very low-mass field dwarf binary resolved in the near-infrared by the Keck II Telescope using laser guide star adaptive optics. Physical association is deduced from the angular proximity of the components and constraints on their common proper motion. We have obtained a near-infrared spectrum of the binary and find that it is best described by an L5+/-0.5 primary and an L7.5+/-0.5 secondary. Model-dependent masses predict that the two components straddle the hydrogen burning limit threshold with the primary likely stellar and the secondary likely substellar. The properties of this sytem - close projected separation (1.8+/-0.3 AU) and near unity mass ratio - are consistent with previous results for very low-mass field binaries. The relatively short estimated orbital period of this system (~7-12 yr) makes it a good target for dynamical mass measurements. Interestingly, the system's angular separation is the tightest yet for any very low-mass binary published from a ground-based telescope and is the tightest binary discovered with laser guide star adaptive optics to date.Comment: 10 pages, 3 figures; accepted for publication to A

A multi-component model of the dynamics of salt-induced hypertension in Dahl-S rats

Background. In humans, salt intake has been suggested to influence blood pressure (BP) on a wide range of time scales ranging from several hours or days to many months or years. Detailed time course data collected in the Dahl salt-sensitive rat strain suggest that the development of salt-induced hypertension may consist of several distinct phases or components that differ in their timing and reversibility. To better understand these components, the present study sought to model the dynamics of salt-induced hypertension in the Dahl salt sensitive (Dahl-S) rat using 3 sets of time course data. Results. The first component of the model ("Acute-Reversible") consisted of a linear transfer function to account for the rapid and reversible effects of salt on BP (ie. acute salt sensitivity, corresponding with a depressed slope of the chronic pressure natriuresis relationship). For the second component ("Progressive-Irreversible"), an integrator function was used to represent the relatively slow, progressive, and irreversible effect of high salt intake on BP (corresponding with a progressive salt-induced shift of the chronic pressure natriuresis relationship to higher BP levels). A third component ("Progressive-Reversible") consisted of an effect of high salt intake to progressively increase the acute salt-sensitivity of BP (ie. reduce the slope of the chronic pressure natriuresis relationship), amounting to a slow and progressive, yet reversible, component of salt-induced hypertension. While the 3 component model was limited in its ability to follow the BP response to rapid and/or brief transitions in salt intake, it was able to accurately follow the slower steady state components of salt-induced BP changes. This model exhibited low values of mean absolute error (1.92 0.23, 2.13 0.37, 2.03 0.3 mmHg for data sets 1 - 3), and its overall performance was significantly improved over that of an initial model having only 2 components. The 3 component model performed well when applied to data from hybrids of Dahl salt sensitive and Dahl salt resistant rats in which salt sensitivity varied greatly in its extent and character (mean absolute error = 1.11 0.08 mmHg). Conclusion. Our results suggest that the slow process of development of salt-induced hypertension in Dahl-S rats over a period of many weeks can be well represented by a combination of three components that differ in their timing, reversibility, and their associated effect on the chronic pressure natriuresis relationship. These components are important to distinguish since each may represent a unique set of underlying mechanisms of salt-induced hypertension

Self-trapping at the liquid vapor critical point

Experiments suggest that localization via self-trapping plays a central role in the behavior of equilibrated low mass particles in both liquids and in supercritical fluids. In the latter case, the behavior is dominated by the liquid-vapor critical point which is difficult to probe, both experimentally and theoretically. Here, for the first time, we present the results of path-integral computations of the characteristics of a self-trapped particle at the critical point of a Lennard-Jones fluid for a positive particle-atom scattering length. We investigate the influence of the range of the particle-atom interaction on trapping properties, and the pick-off decay rate for the case where the particle is ortho-positronium.Comment: 12 pages, 3 figures, revtex4 preprin

Pretransitional phenomena in dilute crystals with first-order phase transition

Pretransitional phenomena at first-order phase transition in crystals diluted by 'neutral' impurities (analogue of nonmagnetic atoms in dilute magnets) are considered. It is shown that field dependence of order parameter becomes nonanalytical in the stability region of the ordered phase, while smeared jumps of thermodynamic parameters and anomalous (non-exponential) relaxation appear near transition temperature of pure crystal.Comment: 4 page

Hydrodynamic singularities and clustering in a freely cooling inelastic gas

We employ hydrodynamic equations to follow the clustering instability of a freely cooling dilute gas of inelastically colliding spheres into a well-developed nonlinear regime. We simplify the problem by dealing with a one-dimensional coarse-grained flow. We observe that at a late stage of the instability the shear stress becomes negligibly small, and the gas flows solely by inertia. As a result the flow formally develops a finite time singularity, as the velocity gradient and the gas density diverge at some location. We argue that flow by inertia represents a generic intermediate asymptotic of unstable free cooling of dilute inelastic gases.Comment: 4 pages, 4 figure