Reflexivity of the translation-dilation algebras on L^2(R)

The hyperbolic algebra A_h, studied recently by Katavolos and Power, is the weak star closed operator algebra on L^2(R) generated by H^\infty(R), as multiplication operators, and by the dilation operators V_t, t \geq 0, given by V_t f(x) = e^{t/2} f(e^t x). We show that A_h is a reflexive operator algebra and that the four dimensional manifold Lat A_h (with the natural topology) is the reflexive hull of a natural two dimensional subspace.Comment: 10 pages, no figures To appear in the International Journal of Mathematic

Scaling Symmetries of Scatterers of Classical Zero-Point Radiation

Classical radiation equilibrium (the blackbody problem) is investigated by the use of an analogy. Scaling symmetries are noted for systems of classical charged particles moving in circular orbits in central potentials V(r)=-k/r^n when the particles are held in uniform circular motion against radiative collapse by a circularly polarized incident plane wave. Only in the case of a Coulomb potential n=1 with fixed charge e is there a unique scale-invariant spectrum of radiation versus frequency (analogous to zero-point radiation) obtained from the stable scattering arrangement. These results suggest that non-electromagnetic potentials are not appropriate for discussions of classical radiation equilibrium.Comment: 13 page

Relating imperatives to action

The aim of this chapter is to provide an analysis of the use of logically complex imperatives, in particular, imperatives of the form Do A1 or A2 and Do A, if B. We argue for an analysis of imperatives in terms of classical logic which takes into account the influence of background information on imperatives. We show that by doing so one can avoid some counter-intuitive results which have been associated with analyses of imperatives in terms of classical logic. In particular, I address Hamblin's observations concerning rule-like imperatives and Ross' Paradox. The analysis is carried out within an agent-based logical framework. This analysis explicates what it means for an agent to have a successful policy for action with respect to satisfying his or her commitments, where some of these commitments have been introduced as a result of imperative language use

Derivation of the Planck Spectrum for Relativistic Classical Scalar Radiation from Thermal Equilibrium in an Accelerating Frame

The Planck spectrum of thermal scalar radiation is derived suggestively within classical physics by the use of an accelerating coordinate frame. The derivation has an analogue in Boltzmann's derivation of the Maxwell velocity distribution for thermal particle velocities by considering the thermal equilibrium of noninteracting particles in a uniform gravitational field. For the case of radiation, the gravitational field is provided by the acceleration of a Rindler frame through Minkowski spacetime. Classical zero-point radiation and relativistic physics enter in an essential way in the derivation which is based upon the behavior of free radiation fields and the assumption that the field correlation functions contain but a single correlation time in thermal equilibrium. The work has connections with the thermal effects of acceleration found in relativistic quantum field theory.Comment: 23 page

The Dipole Coupling of Atoms and Light in Gravitational Fields

The dipole coupling term between a system of N particles with total charge zero and the electromagnetic field is derived in the presence of a weak gravitational field. It is shown that the form of the coupling remains the same as in flat space-time if it is written with respect to the proper time of the observer and to the measurable field components. Some remarks concerning the connection between the minimal and the dipole coupling are given.Comment: 10 pages, LaTe

Rotation of electromagnetic fields and the nature of optical angular momentum

The association of spin and orbital angular momenta of light with its polarization and helical phase fronts is now well established. The problems in linking this with electromagnetic theory, as expressed in Maxwell's equations, are rather less well known. We present a simple analysis of the problems involved in defining spin and orbital angular momenta for electromagnetic fields and discuss some of the remaining challenges. Crucial to our investigation is the duplex symmetry between the electric and magnetic fields

Dark Matter Halo Profiles in Scale-Free Cosmologies

We explore the dependence of the central logarithmic slope of dark matter halo density profiles $\alpha$ on the spectral index $n$ of the linear matter power spectrum $P(k)$ using cosmological $N$-body simulations of scale-free models (i.e. $P(k) \propto k^n$). For each of our simulations we identify samples of well resolved haloes in dynamical equilibrium and we analyse their mass profiles. By parameterising the mass profile using a ``generalised'' Navarro, Frenk & White profile in which the central logarithmic slope $\alpha$ is allowed to vary while preserving the $r^{-3}$ asymptotic form at large radii, we obtain preferred central slopes for haloes in each of our models. There is a strong correlation between $\alpha$ and $n$, such that $\alpha$ becomes shallower as $n$ becomes steeper. However, if we normalise our mass profiles by $r_{-2}$, the radius at which the logarithmic slope of the density profile is -2, we find that these differences are no longer present. We conclude that there is no evidence for convergence to a unique central asymptotic slope, at least on the scales that we can resolve.Comment: 9 pages, 4 figures. Accepted for publication in MNRA

Derivation of the Blackbody Radiation Spectrum from a Natural Maximum-Entropy Principle Involving Casimir Energies and Zero-Point Radiation

By numerical calculation, the Planck spectrum with zero-point radiation is shown to satisfy a natural maximum-entropy principle whereas alternative choices of spectra do not. Specifically, if we consider a set of conducting-walled boxes, each with a partition placed at a different location in the box, so that across the collection of boxes the partitions are uniformly spaced across the volume, then the Planck spectrum correspond to that spectrum of random radiation (having constant energy kT per normal mode at low frequencies and zero-point energy (1/2)hw per normal mode at high frequencies) which gives maximum uniformity across the collection of boxes for the radiation energy per box. The analysis involves Casimir energies and zero-point radiation which do not usually appear in thermodynamic analyses. For simplicity, the analysis is presented for waves in one space dimension.Comment: 11 page

Chaotic dynamics of cold atoms in far-off-resonant donut beam

We describe the classical two dimensinal nonlinear dynamics of cold atoms in far-off-resonant donut beams. We show that there chaotic dynamics exists for charge greater than unity, when the intensity of the beam is periodically modulated. The two dimensional distributions of atoms in $(x,y)$ plane for charge two are simulated. We show that the atoms will acumulate on several ring regions when the system enters to regime of global chaos.Comment: 8 pages, 8 figure