The Schrodinger-like Equation for a Nonrelativistic Electron in a Photon Field of Arbitrary Intensity

The ordinary Schrodinger equation with minimal coupling for a nonrelativistic electron interacting with a single-mode photon field is not satisfied by the nonrelativistic limit of the exact solutions to the corresponding Dirac equation. A Schrodinger-like equation valid for arbitrary photon intensity is derived from the Dirac equation without the weak-field assumption. The "eigenvalue" in the new equation is an operator in a Cartan subalgebra. An approximation consistent with the nonrelativistic energy level derived from its relativistic value replaces the "eigenvalue" operator by an ordinary number, recovering the ordinary Schrodinger eigenvalue equation used in the formal scattering formalism. The Schrodinger-like equation for the multimode case is also presented.Comment: Tex file, 13 pages, no figur

Galaxy growth in the concordance Λ\LambdaCDM cosmology

We use galaxy and dark halo data from the public database for the Millennium Simulation to study the growth of galaxies in the De Lucia et al. (2006) model for galaxy formation. Previous work has shown this model to reproduce many aspects of the systematic properties and the clustering of real galaxies, both in the nearby universe and at high redshift. It assumes the stellar masses of galaxies to increase through three processes, major mergers, the accretion of smaller satellite systems, and star formation. We show the relative importance of these three modes to be a strong function of stellar mass and of redshift. Galaxy growth through major mergers depends strongly on stellar mass, but only weakly on redshift. Except for massive systems, minor mergers contribute more to galaxy growth than major mergers at all redshifts and at all stellar masses. For galaxies significantly less massive than the Milky Way, star formation dominates the growth at all epochs. For galaxies significantly more massive than the Milky Way, growth through mergers is the dominant process at all epochs. At a stellar mass of $6\times 10^{10}M_\odot$, star formation dominates at $z>1$ and mergers at later times. At every stellar mass, the growth rates through star formation increase rapidly with increasing redshift. Specific star formation rates are a decreasing function of stellar mass not only at $z=0$ but also at all higher redshifts. For comparison, we carry out a similar analysis of the growth of dark matter halos. In contrast to the galaxies, growth rates depend strongly on redshift, but only weakly on mass. They agree qualitatively with analytic predictions for halo growth.Comment: 11 pages, 6 figure

On the least common multiple of qq-binomial coefficients

In this paper, we prove the following identity \lcm({n\brack 0}_q,{n\brack 1}_q,...,{n\brack n}_q) =\frac{\lcm([1]_q,[2]_q,...,[n+1]_q)}{[n+1]_q}, where ${n\brack k}_q$ denotes the $q$-binomial coefficient and $[n]_q=\frac{1-q^n}{1-q}$. This result is a $q$-analogue of an identity of Farhi [Amer. Math. Monthly, November (2009)].Comment: 5 page

Multiscale time series modelling with an application to the relativistic electron intensity at the geosynchronous orbit

In this paper, a Bayesian system identification approach to multiscale time series modelling is proposed, where multiscale means that the output of the system is observed at one(coarse) resolution while the input of the system is observed at another (One) resolution. The proposed method identifies linear models at different levels of resolution where the link between the two resolutions is realised via non-overlapping averaging process. This averaged time series at the coarse level of resolution is assumed to be a set of observations from an implied process so that the implied process and the output of the system result in an errors-in-variables ARMAX model at the coarse level of resolution. By using a Bayesian inference and Markov Chain Monte Carlo (MCMC) method, such a modelling framework results in different dynamical models at different levels of resolution at the same time. The new method is also shown to have the ability to combine information across different levels of resolution. An application to the analysis of the relativistic electron intensity at the geosynchronous orbit is used to illustrate the new method