Estimating a bivariate linear relationship

Solutions of the bivariate, linear errors-in-variables estimation problem with unspecified errors are expected to be invariant under interchange and scaling of the coordinates. The appealing model of normally distributed true values and errors is unidentified without additional information. I propose a prior density that incorporates the fact that the slope and variance parameters together determine the covariance matrix of the unobserved true values but is otherwise diffuse. The marginal posterior density of the slope is invariant to interchange and scaling of the coordinates and depends on the data only through the sample correlation coefficient and ratio of standard deviations. It covers the interval between the two ordinary least squares estimates but diminishes rapidly outside of it. I introduce the R package leiv for computing the posterior density, and I apply it to examples in astronomy and method comparison.Comment: 27 pages, 7 figure

Particle creation and particle number in an expanding universe

I describe the logical basis of the method that I developed in 1962 and 1963 to define a quantum operator corresponding to the observable particle number of a quantized free scalar field in a spatially-flat isotropically expanding (and/or contracting) universe. This work also showed for the first time that particles were created from the vacuum by the curved space-time of an expanding spatially-flat FLRW universe. The same process is responsible for creating the nearly scale-invariant spectrum of quantized perturbations of the inflaton scalar field during the inflationary stage of the expansion of the universe. I explain how the method that I used to obtain the observable particle number operator involved adiabatic invariance of the particle number (hence, the name adiabatic regularization) and the quantum theory of measurement of particle number in an expanding universe. I also show how I was led in a surprising way, to the discovery in 1964 that there would be no particle creation by these spatially-flat FLRW universes for free fields of any integer or half-integer spin satisfying field equations that are invariant under conformal transformations of the metric. The methods I used to define adiabatic regularization for particle number, were based on generally-covariant concepts like adiabatic invariance and measurement that were fundamental and determined results that were unique to each given adiabatic order.Comment: 22 pages, no figures, submitted 7May2012 to J. Phys. A for a special issue honoring Prof. Stuart Dowke

Nonlocal theory of area-varying waves on axisymmetric vortex tubes

Area and axial flow variations on rectilinear vortex tubes are considered. The state of the flow is characterized by two dependent variables, a core area, and an azimuthal circulation per unit length, which vary in time and in distance along the length of the tube. Nonlinear integrodifferential equations of motion for these variables are derived by taking certain integrals of the vorticity transport equation. The equations are argued to be valid for moderately short waves (on the order of a few core radii) as well as for long waves. Applications to vortex breakdown and other wave phenomena are considered