Generalized Deformed su(2) Algebras, Deformed Parafermionic Oscillators and Finite W Algebras

Several physical systems (two identical particles in two dimensions, isotropic oscillator and Kepler system in a 2-dim curved space) and mathematical structures (quadratic algebra QH(3), finite W algebra $\bar {\rm W}_0$) are shown to posses the structure of a generalized deformed su(2) algebra, the representation theory of which is known. Furthermore, the generalized deformed parafermionic oscillator is identified with the algebra of several physical systems (isotropic oscillator and Kepler system in 2-dim curved space, Fokas--Lagerstrom, Smorodinsky--Winternitz and Holt potentials) and mathematical constructions (generalized deformed su(2) algebra, finite W algebras $\bar {\rm W}_0$ and W$_3^{(2)}$). The fact that the Holt potential is characterized by the W$_3^{(2)}$ symmetry is obtained as a by-product.Comment: LaTeX, 17 page

Searching for cluster substructure using APM and ROSAT data

We present a detailed study of the morphological features of 22 rich galaxy clusters. Our sample is constructed from a cross-correlation of optical (Abell+APM) data with X-ray (0.1 - 2.4) keV ROSAT pointed observations. We systematically compare cluster images and morphological parameters in an attempt to reliably identify possible substructure in both optical and the X-ray images. To this end, we compute various moments of the optical and X-ray surface-brightness distribution such as the ellipticities, center-of-mass shifts and ellipsoidal orientations. We assess the significance of our results using Monte Carlo simulations. We find significant correlations between the optical and X-ray morphological parameters, indicating that in both parts of the spectrum it is possible to identify correctly the dynamical state of a cluster. Most of our clusters (17/22) have a good 1-to-1 correspondence between the optical and the X-ray images and about 10 appear to have strong indications of substructure. This corresponds to a minimum percentage of order 45 per cent which is in very good accordance with other similar analyses. Finally, 5 out of 22 systems seem to have distinct subclumps in the optical which are not verified in the X-ray images, and thus are suspect of being due to optical projection effects. These results will serve as a useful guide in interpreting subsequent analyses of large optical cluster catalogues.Comment: 15 pages, including 9 figures, MNRAS in press, revised versio