Weak equivalence and non-classifiability of measure preserving actions

Ab\'ert-Weiss have shown that the Bernoulli shift s of a countably infinite group \Gamma is weakly contained in any free measure preserving action (mpa) b of \Gamma. We establish a strong version of this result, conjectured by Ioana, by showing that s \times b is weakly equivalent to b. This is generalized to non-free mpa's using random Bernoulli shifts. The result for free mpa's is used to show that isomorphism on the weak equivalence class of a free mpa does not admit classification by countable structures. This provides a negative answer to a question of Ab\'ert and Elek. We also answer a question of Kechris regarding two ergodic theoretic properties of residually finite groups. An infinite residually finite group \Gamma is said to have EMD if the action p of \Gamma on its profinite completion weakly contains all ergodic mpa's of \Gamma, and \Gamma is said to have property MD if i \times p weakly contains all mpa's of \Gamma, where i denotes the trivial action on a standard non-atomic probability space. Kechris asks if these two properties equivalent and we provide a positive answer by studying the relationship between convexity and weak containment.Comment: 41 pages. This version has minor corrections and updates, including updated reference

Voids in the LCRS versus CDM Models

We have analyzed the distribution of void sizes in the two-dimensional slices of the Las Campanas Redshift Survey (LCRS). Fourteen volume-limited subsamples were extracted from the six slices to cover a large part of the survey and to test the robustness of the results against cosmic variance. Thirteen samples were randomly culled to produce homogeneously selected samples. We then studied the relationship between the cumulative area covered by voids and the void size as a property of the void hierarchy. We find that the distribution of void sizes scales with the mean galaxy separation, $\lambda$. In particular, we find that the size of voids covering half of the area is given by D_{med} \approx \lambda + (12\pm3) \h^{-2}Mpc. Next, by employing an environmental density threshold criterion to identify mock galaxies, we were able to extend this analysis to mock samples from dynamical $n$-body simulations of Cold Dark Matter (CDM) models. To reproduce the observed void statistics, overdensity thresholds of $\delta_{th} \approx 0 ... 1$ are necessary. We have compared standard (SCDM), open (OCDM), vacuum energy dominated ($\Lambda$CDM), and broken scale invariant CDM models (BCDM): we find that both the void coverage distribution and the two-point correlation function provide important and complementary information on the large-scale matter distribution. The dependence of the void statistics on the threshold criterion for the mock galaxy indentification shows that the galaxy biasing is more crucial for the void size distribution than are differences between the cosmological models.Comment: 10 pages, 8 eps figures, submitted to MNRA

A Spatio-Temporal Bayesian Network Classifier for Understanding Visual Field Deterioration

Progressive loss of the field of vision is characteristic of a number of eye diseases such as glaucoma which is a leading cause of irreversible blindness in the world. Recently, there has been an explosion in the amount of data being stored on patients who suffer from visual deterioration including field test data, retinal image data and patient demographic data. However, there has been relatively little work in modelling the spatial and temporal relationships common to such data. In this paper we introduce a novel method for classifying Visual Field (VF) data that explicitly models these spatial and temporal relationships. We carry out an analysis of this method and compare it to a number of classifiers from the machine learning and statistical communities. Results are very encouraging showing that our classifiers are comparable to existing statistical models whilst also facilitating the understanding of underlying spatial and temporal relationships within VF data. The results reveal the potential of using such models for knowledge discovery within ophthalmic databases, such as networks reflecting the ‘nasal step’, an early indicator of the onset of glaucoma. The results outlined in this paper pave the way for a substantial program of study involving many other spatial and temporal datasets, including retinal image and clinical data