Small oscillations and the Heisenberg Lie algebra

The Adler Kostant Symes [A-K-S] scheme is used to describe mechanical systems for quadratic Hamiltonians of $\mathbb R^{2n}$ on coadjoint orbits of the Heisenberg Lie group. The coadjoint orbits are realized in a solvable Lie algebra $\mathfrak g$ that admits an ad-invariant metric. Its quadratic induces the Hamiltonian on the orbits, whose Hamiltonian system is equivalent to that one on $\mathbb R^{2n}$. This system is a Lax pair equation whose solution can be computed with help of the Adjoint representation. For a certain class of functions, the Poisson commutativity on the coadjoint orbits in $\mathfrak g$ is related to the commutativity of a family of derivations of the 2n+1-dimensional Heisenberg Lie algebra $\mathfrak h_n$. Therefore the complete integrability is related to the existence of an n-dimensional abelian subalgebra of certain derivations in $\mathfrak h_n$. For instance, the motion of n-uncoupled harmonic oscillators near an equilibrium position can be described with this setting.Comment: 17 pages, it contains a theory about small oscillations in terms of the AKS schem

Stripe to spot transition in a plant root hair initiation model

A generalised Schnakenberg reaction-diffusion system with source and loss terms and a spatially dependent coefficient of the nonlinear term is studied both numerically and analytically in two spatial dimensions. The system has been proposed as a model of hair initiation in the epidermal cells of plant roots. Specifically the model captures the kinetics of a small G-protein ROP, which can occur in active and inactive forms, and whose activation is believed to be mediated by a gradient of the plant hormone auxin. Here the model is made more realistic with the inclusion of a transverse co-ordinate. Localised stripe-like solutions of active ROP occur for high enough total auxin concentration and lie on a complex bifurcation diagram of single and multi-pulse solutions. Transverse stability computations, confirmed by numerical simulation show that, apart from a boundary stripe, these 1D solutions typically undergo a transverse instability into spots. The spots so formed typically drift and undergo secondary instabilities such as spot replication. A novel 2D numerical continuation analysis is performed that shows the various stable hybrid spot-like states can coexist. The parameter values studied lead to a natural singularly perturbed, so-called semi-strong interaction regime. This scaling enables an analytical explanation of the initial instability, by describing the dispersion relation of a certain non-local eigenvalue problem. The analytical results are found to agree favourably with the numerics. Possible biological implications of the results are discussed.Comment: 28 pages, 44 figure

The energy spectrum observed by the AGASA experiment and the spatial distribution of the sources of ultra-high energy cosmic rays

Seven and a half years of continuous monitoring of giant air showers triggered by ultra high-energy cosmic rays have been recently summarized by the AGASA collaboration. The resulting energy spectrum indicates clearly that the cosmic ray spectrum extends well beyond the Greisen-Zatsepin-Kuzmin (GZK) cut-off at $\sim 5 \times 10^{19}$ eV. Furthermore, despite the small number statistics involved, some structure in the spectrum may be emerging. Using numerical simulations, it is demonstrated in the present work that these features are consistent with a spatial distribution of sources that follows the distribution of luminous matter in the local Universe. Therefore, from this point of view, there is no need for a second high-energy component of cosmic rays dominating the spectrum beyond the GZK cut-off.Comment: 14 pages, 4 figures, Astrophys. J. Letters (submitted

On the significance of the observed clustering of ultra-high energy cosmic rays

Three pairs of possibly correlated ultra-high energy cosmic ray events were reported by Hayashida et al (1996). Here we calculate the propagation of the corresponding particles through both the intergalactic and galactic magnetic fields. The large scale disc and halo magnetic components are approximated by the models of Stanev (1997). The intergalactic magnetic field intensity is modulated by the actual density of luminous matter along the corresponding lines of sight, calculated from the CfA redshift catalogue (Huchra et al, 1995). The results indicate that, if the events of each pair had a common source and were simultaneously produced, they either originated inside the galactic halo or otherwise very unlikely events were observed. On the other hand, an estimate of the arrival probability of ultra-high energy cosmic rays, under the assumption that the distribution of luminous matter in the nearby universe traces the distribution of the sources of the particles and intensity of the intergalactic magnetic field, suggests that the pairs are chance clusterings.Comment: Ap. J. Letters Accepted - 13 pages + 4 figure

Magneto-Conductance Anisotropy and Interference Effects in Variable Range Hopping

We investigate the magneto-conductance (MC) anisotropy in the variable range hopping regime, caused by quantum interference effects in three dimensions. When no spin-orbit scattering is included, there is an increase in the localization length (as in two dimensions), producing a large positive MC. By contrast, with spin-orbit scattering present, there is no change in the localization length, and only a small increase in the overall tunneling amplitude. The numerical data for small magnetic fields $B$, and hopping lengths $t$, can be collapsed by using scaling variables $B_\perp t^{3/2}$, and $B_\parallel t$ in the perpendicular and parallel field orientations respectively. This is in agreement with the flux through a `cigar'--shaped region with a diffusive transverse dimension proportional to $\sqrt{t}$. If a single hop dominates the conductivity of the sample, this leads to a characteristic orientational `finger print' for the MC anisotropy. However, we estimate that many hops contribute to conductivity of typical samples, and thus averaging over critical hop orientations renders the bulk sample isotropic, as seen experimentally. Anisotropy appears for thin films, when the length of the hop is comparable to the thickness. The hops are then restricted to align with the sample plane, leading to different MC behaviors parallel and perpendicular to it, even after averaging over many hops. We predict the variations of such anisotropy with both the hop size and the magnetic field strength. An orientational bias produced by strong electric fields will also lead to MC anisotropy.Comment: 24 pages, RevTex, 9 postscript figures uuencoded Submitted to PR

Acceptance of fluorescence detectors and its implication in energy spectrum inference at the highest energies

Along the years HiRes and AGASA experiments have explored the fluorescence and the ground array experimental techniques to measure extensive air showers, being both essential to investigate the ultra-high energy cosmic rays. However, such Collaborations have published contradictory energy spectra for energies above the GZK cut-off. In this article, we investigate the acceptance of fluorescence telescopes to different primary particles at the highest energies. Using CORSIKA and CONEX shower simulations without and with the new pre-showering scheme, which allows photons to interact in the Earth magnetic field, we estimate the aperture of the HiRes-I telescope for gammas, iron nuclei and protons primaries as a function of the number of simulated events and primary energy. We also investigate the possibility that systematic differences in shower development for hadrons and gammas could mask or distort vital features of the cosmic ray energy spectrum at energies above the photo-pion production threshold. The impact of these effects on the true acceptance of a fluorescence detector is analyzed in the context of top-down production models