A continuous movement version of the Banach—Tarski paradox: A solution to de Groot's Problem

In 1924 Banach and Tarski demonstrated the existence of a paradoxical decomposition of the 3-ball B, i.e., a piecewise isometry from B onto two copies of B. This article answers a question of de Groot from 1958 by showing that there is a paradoxical decomposition of B in which the pieces move continuously while remaining disjoint to yield two copies of B. More generally, we show that if n ≥ 2, any two bounded sets i

The effect of small inter-pulsar distance variations in stochastic gravitational wave background searches with Pulsar Timing Arrays

One of the primary objectives for Pulsar Timing Arrays (PTAs) is to detect a stochastic background generated by the incoherent superposition of gravitational waves (GWs), in particular from the cosmic population of supermassive black hole binaries. Current stochastic background searches assume that pulsars in a PTA are separated from each other and the Earth by many GW wavelengths. As more millisecond pulsars are discovered and added to PTAs, some may be separated by only a few radiation wavelengths or less, resulting in correlated GW phase changes between close pulsars in the array. Here we investigate how PTA overlap reduction functions (ORFs), up to quadrupole order, are affected by these additional correlated phase changes, and how they are in turn affected by relaxing the assumption that all pulsars are equidistant from the solar system barycenter. We find that in the low frequency GW background limit of $f\sim10^{-9}$~Hz, and for pulsars at varying distances from the Earth, that these additional correlations only affect the ORFs by a few percent for pulsar pairs at large angular separations, as expected. However when nearby (order 100 pc) pulsars are separated by less than a few degrees, the correlated phase changes can introduce variations of a few tens of percent in the magnitude of the isotropic ORF, and much larger fractional differences in the anisotropic ORFs-- up to 188 in the $m=0$, $l=2$ ORF for equidistant pulsars separated by 3 degrees. In fact, the magnitude of most of the anisotropic ORFs is largest at small, but non-zero, pulsar separations. Finally, we write down a small angle approximation for the correlated phase changes which can easily be implemented in search pipelines, and for completeness, examine the behavior of the ORFs for pulsars which lie at a radiation wavelength from the Earth.Comment: 16 pages, 8 figures, submitted to PR

Rational and Coalition Models of Job Evaluation: Do More Powerful University Departments Have an Advantage?

Job evaluation research has, to date, focused on the individual as the unit of analysis. After approximately 50 years of study, evidence on the basic assumptions supporting job evaluation is still inconclusive. This study expands the research by employing organizational theory to the topic and studying job evaluation at the group level. Prior work on rational and coalition models of resource allocation is used to develop hypotheses that are tested with six years of job evaluation data from a university. The results support the coalition model and the conclusion that departmental power can affect job evaluation outcomes

Galaxies in Clusters: the Observational Characteristics of Bow-Shocks, Wakes and Tails

The dynamical signatures of the interaction between galaxies in clusters and the intracluster medium (ICM) can potentially yield significant information about the structure and dynamical history of clusters. To develop our understanding of this phenomenon we present results from numerical modelling of the galaxy/ICM interaction, as the galaxy moves through the cluster. The simulations have been performed for a broad range, of ICM temperatures (kT = 1,4 and 8 keV), representative of poor clusters or groups through to rich clusters. There are several dynamical features that can be identified in these simulations; for supersonic galaxy motion, a leading bow-shock is present, and also a weak gravitationally focussed wake or tail behind the galaxy (analogous to Bondi-Hoyle accretion). For galaxies with higher mass-replenishment rates and a denser interstellar medium (ISM), the dominant feature is a dense ram-pressure stripped tail. In line with other simulations, we find that the ICM/galaxy ISM interaction can result in complex time- dependent dynamics, with ram-pressure stripping occurring in an episodic manner. In order to facilitate this comparison between the observational consequences of dynamical studies and X-ray observations we have calculated synthetic X-ray flux and hardness maps from these simulations. These calculations predict that the ram-pressure stripped tail will usually be the most visible feature, though in nearby galaxies the bow-shock preceding the galaxy should also be apparent in deeper X-ray observations. We briefly discuss these results and compare with X-ray observations of galaxies where there is evidence of such interactions.Comment: 14 pages, 8 diagrams, MNRAS (in press