In this paper, we describe a representation for spatial information, called the stochastic map, and associated procedures for building it, reading information from it, and revising it incrementally as new information is obtained. The map contains the
estimates of relationships among objects in the map, and their uncertainties, given all the available information. The procedures provide a general solution to the problem of estimating uncertain relative spatial relationships. The estimates are probabilistic in nature, an advance over the previous, very conservative, worst-case approaches to the problem. Finally, the procedures are developed in the context of state-estimation and filtering theory, which provides a solid basis for numerous extensions.
We examine high signal to noise XMM-Newton European Photon Imaging Camera (EPIC) and Reflection Grating Spectrometer (RGS) observations to determine the physical characteristics of the gas in the cool core and outskirts of the nearby rich cluster A31
12. The XMM-Newton Extended Source Analysis Software data reduction and background modeling methods were used to analyze the XMM- Newton EPIC data. From the EPIC data we find that the iron and silicon abundance gradients show significant increase towards the center of the cluster while the oxygen abundance profile is centrally peaked but has a shallower distribution than that of iron. The X-ray mass modeling is based on the temperature and deprojected density distributions of the intra-cluster medium determined from EPIC observations. The total mass of A3112 obeys the M-T scaling relations found using XMM-Newton and Chandra observations of massive clusters at R500. The gas mass fraction f_gas= 0.149^{+0.036}_{-0.032} at R500, is consistent with the seven-year WMAP results. The comparisons of line fluxes and flux limits on the Fe XVII and Fe XVIII lines obtained from high resolution RGS spectra indicate that there is no spectral evidence for cooler gas associated with the cluster with temperature below 1.0 keV in the central <38 (sim 52 kpc) region of A3112. High resolution RGS spectra also yield an upper limit to the turbulent motions in compact core of A3112 (206 km/s). We find that the energy contribution of turbulence to total energy is less than 6 per cent. This upper limit is consistent with the amount of energy contribution measured in recent high resolution simulations of relaxed galaxy clusters.
